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If we were to measure loudness solely as a property of amplitude, then high frequency sounds, which contain high energy, would have the same value as low frequency sounds, which contain low energy. On the other hand, if we were to attempt to measure loudness solely as a property of energy, then we find that "much more acoustic energy is required in the lower frequencies to create sounds of equal loudness to those in higher frequencies" (Citation).

What function of amplitude and frequency has the best linear correlation with perceived loudness, for an average speaking or music volume? Or put another way, if we specify amplitude = frequency^k, then what value k best matches the equal-loudness curves?

https://en.wikipedia.org/wiki/Fletcher%E2%80%93Munson_curves

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I'm not entirely well versed in the mathematics behind acoustics, but I don't think you can map a purely linear correlation with perceived loudness. There are multiple kinds of frequency functions related to loudness, and none of them are perfect but are rather based on perceptual biases.

I'd recommend giving this webpage (and the whole website really) about loudness: Loudness Explained

These guys do a really good job of explaining the current loudness standards being used for Broadcast in a variety of world regions and what the differences are in the standards between regions.

I think what you're specifically referring to may fall under the category of K-Weighting.

Many recording devices now come with K-Weighting options and many loudness meter and frequency analyzer plugins offer K-Weighting options so you can view your amplitude and/or frequency distribution based on specific weighted biases.

Again, academically I'm not well versed in the mathematics you're referring to, but I can tell you from a creative standpoint, you should always trust your ears, regardless of the math!

  • If it is K-Weighting that we're talking about, you'll find that most weighting options are NOT based on linear correlations, but something non-linear and not quite exponential either, it's more based on average human HRTFs. ¯_(ツ)_/¯ Sorry if that wasn't helpful. – user2188400 May 22 '17 at 19:18

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