So, everyone knows the equal-loudness curves, which show the sound pressure (in dB SPL) needed to achieve a particular loudness in phons through different frequencies.

This curve is fundamental for many areas of acoustics and hearing, however, I wanted to bring more information out of it.

Firstly, they don't show the variation in loudness of sounds with a same sound pressure through different frequencies. That is very important, specially if you want to understand how a "flat response" should sound like when you feed it with pure tones of the same amplitude.

Secondly, the unit phon itself isn't the closest to be linear with perceived loudness. The unit sone was made for that, but needs corrections for very loud and very quiet sound (and it's a bit of an approximation anyway, loudness is very subjective, but this is still better for a psychophysical approach than phon).

So, I took the data from ISO 226, converting the values from phons to sones and the values from db SPL to Pa (pascals) to find a loudness curve for

For example, for the 100 Hz frequency, the data is:

I then performed a fitting to find the values for other sound pressures. The model I used is a simple power-law expression:

L = A·p

Where L is the loudness, A is a constant, p is the sound pressure and γ the exponent (a dimensionless constant)

Not only this model provided a near-perfect fit (R² > 0.999 always), it is also the same used for gamma correction, and the value of γ (gamma) tells a lot about dynamics.

If

If

If

The constant A is a different one for each frequency, and also has a different dimension for each. Its unit has to be sone·Pa

Here's a graph of how the A constants and γ values vary with frequency

The graph on the left will give the overall shape of the equal-pressure loudness contours. The graph on the right tells about the dynamics of each frequency and how the contours will change with different sound pressures.

So, using these data, it's possible to make the contours I wished for.

Here they are:

Each curve in the graph on the left shows the variation in perceived loudness for pure tones of different frequencies with the same sound pressure.

The curves on the graph on the right has been normalized so the loudest frequency of each curve has a value 1, and it helps visualizing how the proportions between the loudness of each frequency changes with sound pressure (due to their different dynamics).

In the curve for 20 mPa (the cyan curve), the frequencies around 15 kHz are just about 40% as loud as the frequencies around 3 kHz (the loudest in all cases). But, for 2 Pa (black curve), they're 80% as loud as 3 kHz. Their loudness relative to 3 kHz double.

The bass frequencies change even more. In this same range (from 20 mPa to 2 Pa), frequencies around 40 Hz go from being 1% as loud as 3 kHz to 10%.

The frequencies between 2 and 6 kHz keep their proportional loudness because their γ are nearly the same, around 0.585 (they have the same dynamics).

Although the data may be imprecise, this is just an approximation when it comes to your actual loudness perception. But in a way or the other, this is still useful for a ton of qualitative observations.

The main observation being that, no mater how an equipment is good at keeping its profile (flat or whatever other it has), the equalization will sound different at different SPL, increasing the overall loudness will also increase the bass and mid-bass (from 20 to 600 Hz) and treble (12-17 kHz) relative to mid-highs(2-6 kHz).

These contours also help "equalizing by ear", as they give a notion of how it should sound, taking the loudness of 3 kHz as a reference. The only thing you need to know then it's the sound pressure level you're dealing with.

For a reference, here's a little table I took from wikipedia:

This curve is fundamental for many areas of acoustics and hearing, however, I wanted to bring more information out of it.

Firstly, they don't show the variation in loudness of sounds with a same sound pressure through different frequencies. That is very important, specially if you want to understand how a "flat response" should sound like when you feed it with pure tones of the same amplitude.

Secondly, the unit phon itself isn't the closest to be linear with perceived loudness. The unit sone was made for that, but needs corrections for very loud and very quiet sound (and it's a bit of an approximation anyway, loudness is very subjective, but this is still better for a psychophysical approach than phon).

So, I took the data from ISO 226, converting the values from phons to sones and the values from db SPL to Pa (pascals) to find a loudness curve for

**each frequency,**relating the perceived loudness with sound pressure.For example, for the 100 Hz frequency, the data is:

I then performed a fitting to find the values for other sound pressures. The model I used is a simple power-law expression:

L = A·p

^{γ}Where L is the loudness, A is a constant, p is the sound pressure and γ the exponent (a dimensionless constant)

Not only this model provided a near-perfect fit (R² > 0.999 always), it is also the same used for gamma correction, and the value of γ (gamma) tells a lot about dynamics.

If

**γ = 1**for a given frequency, then the perceived loudness is**directly proportional**to the sound pressure. Which only happens in one frequency, around 60 Hz.If

**γ < 1**, then the perceived loudness is**compressed**, meaning that doubling the sound pressure doesn't double the perceived loudness. Everything above 60 Hz is compressed, and the most compressed frequencies are exactly those we are most sensitive to, between 2 and 6 kHz.If

**γ > 1**, which is the case for everything below 60 Hz, then the perceived loudness is**expanded**, so doubling the sound pressure more-than-doubles the perceived loudness.The constant A is a different one for each frequency, and also has a different dimension for each. Its unit has to be sone·Pa

^{-γ}, so it's only the same dimension if γ is the same. Nevertheless, it's still useful to compare their values, for they tell a lot about how each frequency is "intrinsically loud".Here's a graph of how the A constants and γ values vary with frequency

The graph on the left will give the overall shape of the equal-pressure loudness contours. The graph on the right tells about the dynamics of each frequency and how the contours will change with different sound pressures.

So, using these data, it's possible to make the contours I wished for.

Here they are:

Each curve in the graph on the left shows the variation in perceived loudness for pure tones of different frequencies with the same sound pressure.

The curves on the graph on the right has been normalized so the loudest frequency of each curve has a value 1, and it helps visualizing how the proportions between the loudness of each frequency changes with sound pressure (due to their different dynamics).

In the curve for 20 mPa (the cyan curve), the frequencies around 15 kHz are just about 40% as loud as the frequencies around 3 kHz (the loudest in all cases). But, for 2 Pa (black curve), they're 80% as loud as 3 kHz. Their loudness relative to 3 kHz double.

The bass frequencies change even more. In this same range (from 20 mPa to 2 Pa), frequencies around 40 Hz go from being 1% as loud as 3 kHz to 10%.

The frequencies between 2 and 6 kHz keep their proportional loudness because their γ are nearly the same, around 0.585 (they have the same dynamics).

Although the data may be imprecise, this is just an approximation when it comes to your actual loudness perception. But in a way or the other, this is still useful for a ton of qualitative observations.

The main observation being that, no mater how an equipment is good at keeping its profile (flat or whatever other it has), the equalization will sound different at different SPL, increasing the overall loudness will also increase the bass and mid-bass (from 20 to 600 Hz) and treble (12-17 kHz) relative to mid-highs(2-6 kHz).

These contours also help "equalizing by ear", as they give a notion of how it should sound, taking the loudness of 3 kHz as a reference. The only thing you need to know then it's the sound pressure level you're dealing with.

For a reference, here's a little table I took from wikipedia:

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