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Decibels are logarithmic with a base of 10. I've been told before that two car horns are not twice as loud as one car horn. Rather, it takes ten car horns to be twice as loud, because of the log10 nature of decibels and our perception of sound.

Why does our perception of sound intensity have a logarithmic base of 10, instead of 6, 1.234, or any other real number greater than 1? That is, why doesn't it take 6 (or any other real number greater than 1) car horns to be twice as loud as one? It seems kind of arbitrary to me that 10 is the magic number.

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  • $\begingroup$ Related: physics.stackexchange.com/q/173185, physics.stackexchange.com/q/6588 and links therein. $\endgroup$
    – Kyle Kanos
    Sep 23, 2015 at 14:18
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    $\begingroup$ I'm voting to close this question as off-topic because it is about human perception of sound not physics $\endgroup$ Sep 23, 2015 at 14:36
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    $\begingroup$ Well, on psychology forums he won't exactly get a mathematical argument why that logarithm has base 10. Also, the problem is more general than just sound. $\endgroup$ Sep 23, 2015 at 14:51
  • $\begingroup$ @JohnRennie I really don't think I'd get a physics-based answer if I asked anywhere else. That's what I came here for. Not sure if there actually is something in physics which answers my question, but I figure it's worth a shot. Maybe it's a personal vendetta against the number 10 (I wish we would all switch to dozenal), but I feel it's very arbitrary. And I don't think the average psychologist or even neuroscientist is qualified to answer. $\endgroup$ Sep 23, 2015 at 14:59
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    $\begingroup$ Note that you can convert the base of logarithmic system to be what every you want without changing the logarithmic character. Bels are a base-10 system because the creators preferred base 10 to base 2 or base $e$. There is neither physics nor biology involved. $\endgroup$ Sep 23, 2015 at 15:49

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The basics lie in Weber-Fechner law. That would suggest generally any kind of your logarithm because the trend and tendency are the key factors. So, ok, you are basically right.

In my opinion, the reasons for 10 are practical:

  • Order estimation and readability of charts with log axes are way easier
  • Other units as phons, sons etc. are defined using dB with 10 factor
  • Signal processing theory etc. uses the 10 as well
  • Some definitions ends on 'nicer' forms, e.g.:

$$ L_I = 10 \log \frac{I}{I_0} $$

for $L$ defined by intensity. But since $I \sim p^2$ and we want $L_I \sim L_p$:

$$ L_p = 20 \log \frac{p}{p_0} $$

which is, you know, better than 2.468.

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  • $\begingroup$ So we use a base of 10 for pragmatism over accuracy? $\endgroup$ Sep 23, 2015 at 16:30
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    $\begingroup$ Well, since Fechner-Weber's law is only the first estimation at its best, I wouldn't be sure about 'accuracy'. There are influences of so many secondary quantities (e.g. pitch for loudness and vice versa) that you really can't be sure of 'accuracy' of these formulas at all. $\endgroup$ Sep 23, 2015 at 16:39
  • $\begingroup$ @MarcusMcLean In fact decibel has nothing to do with what we perceive. The brain's reactions may or may not be linear with regards to neural input, and the impulse generated may also be non-linear with regards to power(W) or magnitude(dB) of sound input. Also everybody's brain may process auditory stimuli differently, so you can't even say "twice as loud" in the strict sense because what appears to be twice as loud to you may not be so to someone else. $\endgroup$
    – busukxuan
    Jan 15, 2016 at 18:23
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Loudness is a subjective measurement made from testing a large population of human subjects and how they perceive different sound intensities. It, like the intensity level, is a logarithmic behavior. Again, one could use any logarithmic base. But ....

Acoustic scientists have done experiments with pure tones (sine waves) and searched for the just noticeable difference (JND) of loudness in sequentially played tones of the same frequency (say, tone A at level $\beta_A$ and tone B at level $\beta_B$, A&B at the same frequency). The required change in level for JND across the audible frequency spectrum is close to 1 dB (using a base 10), varying from .5 dB to 1.5 dB.

Base 10 seems to give us a good description, so it's a serendipitous, pragmatic, useful base. It actually seems to work well with what is happening inside our heads. Plus it makes calculations easy because there are no base 3.14159 logarithm buttons.

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I think it was "just" log-based in shape first, and then the scale invented to make "1" the approximate meaningful change. Making that unit 1/10 of a larger unit instead of the real named thing on its own is a human artifact. He could have called it a Fleeb and then nobody would suppose that 10 Fleebs was anything magical or significant other than being 10 times the basic unit.

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There is no such thing as perceived sound intensity. Humans do not perceive sound intensity which is the product of sound pressure and particle velocity and it is used to model transport of energy in sound fields. Humans only sense sound pressure. The perception is called loudness and it is more complicated than a base 10 logarithm. The statement that it takes ten horns to be twice as loud is a very rough simplification. It might be based on the fact that under certain circumstances a 10 dB increase in sound pressure level is on the average perceived as twice as loud. However, 10 dB may or may not equal nine more horns. Loudness analyzers have been developed to model human hearing and to predict perceptions of sound. Generally these analyzers require sound pressure time series as input meaning that you would also need to model the transmission of sound from the horns to the listener.

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