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# The non-linearities of the Human Ear

## Background

While you do certainly want to pay attention to linearity when investigating audio equipment, it is interesting to realize how intrinsically non-linear human hearing behaves. This page describes the many non-linearities affecting our hearing.

## The logarithmic scale

First, let's explain what the logarithmic scale is. On a logarithmic scale, numbers are evenly spaced, not by an additive factor - this is the linear scale - but a multiplicative factor.

```
Linear Scale        0   1   2   3   4   5   6   7   8   9   10   (+1)
Logarithmic Scale   1   2   4   8  16  32  64  128 256 512 1024  (x2)

```

The linear scale is the most intuitive when it represents numbers along an axis; the logarithmic one is much more powerful when one needs to work with a large dynamic range. Look at the two scales above: by the same amount of space, the linear scale only goes up to 10, while the logarithmic extends to 1024! And remarkably, the logarithmic scale keeps the same precision as the linear scale around the small numbers. Most human perception works that way. It allows us to discern subtleties around the smaller inputs, yet respond to the bigger inputs without being overrun. More than this, logarithmic is also the way we often do think, too. Let's take an example.

Money. A \$1,000 salary increase is truly substantial... when you only earn that much. However, the same amount of money would be perceived as marginal, if you were drawing a salary of six figures. Another one: you don't need a quote down to the cent when buying a car, but a few cents may matter when buying a beer. That's exactly what the logarithmic scale does naturally for you. Take a look at the series above again: you can easily mark numbers down to a decimal precision on the beginning of the logarithmic scale (like 1.5 standing right in between 1 and 2) but won't care as much at the other end of the scale: if you were asked to position the numbers 800 and 853, you'd locate them somewhere between 512 and 1024, without much of a difference between the two.

Some readers will correct me by saying that 1 2 4 8 ... is not a logarithmic series, but an exponential one. They are totally right. But logarithmic is the wording that is commonly used. It comes from the fact that it takes a logarithm to turn the exponential series into the linear scale. Also note that sound pressure levels are often expressed in decibels (dB), a unit that is intrinsically logarithmic: a linear scale in dB is exponential in terms of acoustic pressure levels.

## Basically, the ear is logarithmic

So, it shouldn't surprise you that our hearing works on a logarithmic scale too. Let's experience this through a couple of audible examples.

The frequency test, first. Let's take 200 Hz, as a starting point. Our next frequency in the scale will be 300 Hz. From there, we will keep adding 100 Hz in the linear series, or multiplying by 1.5 in the logarithmic series:

```
Linear Series        200  300  400  500  600  700  800  900  1000 1100  (+100)
Logarithmic Series   200  300  450  675  1012 1519 2278 3417 5126 7689  (x1.5)

```
 Linear Frequency Log Frequency Test

Listen to both, and ask yourself which one does sound evenly spaced to you? And the answer will be... the logarithmic series

 Linear Level Log Level Test

This applies with levels too. With the linear scale sound level test, levels increase linearly as follows: 5% 10% 15% 20% 25% of the maximum dynamic range allowed by the sound file (0 dBFS). With the logarithmic test, levels are doubling from one step to the next (+6dB): 5% 10% 20% 40% 80%. Which one sounds the most evenly spaced to you? The logarithmic scale again.

## Frequency Sensitivity

Then comes another effect: the increased sensitivity of the hearing at certain frequencies. The hearing is far from offering a flat frequency response: our sensitivity peaks around 3 kHz. This means that sounds around that frequency will be precieved as louder, even if their sound pressure level is the same.

 300 v/s 3k @ -24dBFS

Our test file alternates two tones, 300 Hz and 3 kHz, both played back at -24 dBFS. Take a listen and you will probably agree that the highest pitch also feels the loudest. In order to achieve the most dramatic difference, you should turn the volume down, and perform the test around your hearing thresholds.

## Variations of the Frequency Sensitivity depending on the level

The last effect is probably the most evil for audio enthusiasts. The difference in frequency sensitivity mentioned above, actually depends on sound levels! At lower levels, our ear sensitivity in the bass and treble areas, drops significantly. That's why the test above worked better at quieter levels. Exposed to loud music, the frequency response of our ears becomes somehow flatter. This effect is described by the so-called Fletcher-Munson curves. They show the sound pressure levels to effectively apply in order to achieve a given perceived level.

In practice, this means that the same music only played quieter will sound duller. This is the reason why the loudness button has been invented: to artificially compensate for the drop lower and upper ends which naturally occur at quieter levels. This is also one of the reasons behind the so-called loudness war, a term that refers to artists and studio engineers trying to burn the loudest levels into a recording: being louder than competition creates the illusion that music sounds better, because of the more prominent bass and treble, which is difficult not to like. Unfortunately, making a recording louder causes problems too, such as decreasing the dynamic range and introducing distortions. But this is another story.

 Original Orig+1dB Orig+3dB

Listen to the original sound file, then the two other versions. One decibel difference is almost impossible to detect, yet it is a well known bias during sound quality blind tests. A three decibel difference is audible, and obvious in our case as we told you what happened to the file. Now imagine that we came with another plausible explanation - a subtle EQ maybe, or the use of better A/D converters - how many of you would NOT have been trapped?

## Conclusions

Human hearing is everything but linear and flat. Simply changing a reference level, will impact the way the sound will be perceived: not only louder or quieter, but with a different tone as well. This puts frequency flatness into perspective. Take headphones, for example. I personally can tolerate a little emphasis on both the bass and treble extremes: the headphones I am using are not perfectly flat. However, this means that I can listen to my source material at lower levels, save my hearing, and have the headphones simulate what I would be hearing at higher levels with flat headphones! Isn't that nice?

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