# Textbook vs real world explanation of loudness perception

I read that if you double the number of instruments playing, i.e double the amount of acoustic energy, then the perceived loudness rises by 3 dB. I've also read that a 3 dB change is almost imperceptible.

However, two trumpets playing at the same time seems considerably louder than just one.

Can someone explain?

At least one of these three statements must be false:

• Double the acoustic power and the sound volume rises by 3 dB
• A 3 dB change in volume is imperceptible
• two trumpets playing exactly the same thing are considerably louder than just one trumpet playing alone.

I wouldn't be surprised if I'd mixed up physics, which is about experiments and actual physical measurements, with psycho-acoustics which is about the perception of sound, and hence not measurable in the same way.

• This question needs some work on rewording. You are citing several sources of information that seem contradictory. Please state where you found this info. Is one a physics book and the other a sound system or recording magazine?
– user50691
Jul 6, 2018 at 12:09
• I suspect you've confused 3dBA, an absolute (and very faint) sound level, with a 3dB change in sound level. Jul 6, 2018 at 13:30

First of all you have to have a technical definition of "loudness" before the question makes sense. I will try to add to this with some "textbook" info and I hope it clarifies things but the question may need modification as more info comes in.

From a purely physics based point of view we define, or associate, the loudness of a set of acoustic sources with amplitude or power of the vibrating sources. Power is proportional to the square of the amplitude. Since sound is actually a wave the amplitudes add producing an interference pattern. In an acoustic environment where there is reverb and background noise we add power to get the total power produced by the sources.

The Decibel scale is based on taking the logarithm of the total power, scaled by a reference power (in music this is usually the threshold of human hearing). The formula is 10*log(P). If you double the power (or more generally the argument of the log) you get ~3.0*(The dB of the original power). This is a purely mathematical measure of the objective relative strength of two equal strength sources. So the first problem in the question is "double the number of players". To conclude that the loudness level is 3dB higher you are implicitly assuming that they are (1) all playing the same strength, (2) that the combination is incoherent, and (3) that you are in an environment where you are not sitting in a place of destructive interference.

The next point is that the human ear and brain work non-linearly. For you to perceive that the situation is 3dB higher (twice the power) the combined music sources need to trigger your audio detectors (ears+brain) to perceive the change. This is partly objective and related to the response curve of the ear, and partly subjective related to the ability of someone to judge what is "loud" relative to a set of linear devices set up to measure power. So it is entirely possible to set up a controlled experiment where a person sits in a room listening to 1, 2, 4, 8, equal sources and doesn't necessarily judge the results as increasing by 3dB each time.

Factors that affect perception of loudness include (1) saturation of of ear and its lack of ability to respond to increased amplitude, (2) placement of the sources (you could be sitting in a dead spot), (3) psycho-acoustic effects, (4) the actual frequency of the instruments. The last one is very interesting. If you produce tones with a scientific device, one low pitch and the other high pitch (say 55Hz, and 1500Hz) both at exactly the same power, most humans will say the bass note is quieter than than treble note. So the second issue with your proposed question is the comparison of "two trumpets" versus "3dB is almost imperceptible". Without knowing the situation the statements about doubling trumpets and 3dB is imperceptible are meaningless. The later could be true when the instruments are not perceptible over background reverb in a room, or if the situation is 2 basses playing in low register at piano or mezzo-piano (not perceptible in the first place, relative loudness is very hard to judge).

So in short scientific measure of loudness and human perception of loudness do not match in general. Even when they should (in frequency bands where most humans perception matches objective measure) environmental factors can cause wild variations in response to the question. Since acoustics is a field that travels through space the "power" at the sources will not equal "power" at the ear.

Edit (addition). Another cause of ambiguity is that the power that reaches your ear will attenuate, 1/R^2 loss. This is geometric spread. So when you "add a trumpet" to the band if one of them is farther way from you than the other the combination will not be twice as powerful at your ear. This is part of the wave nature of sound. To get the correct combination a person needs to weight the powers by the inverse of the distance squared. Even if they blow the same note with equal strength the combined power at the ear will not be 3dB higher. Again, there is a bit of ambiguity in the distillation of the items you have presented.

• "power that reaches your ear will attenuate, 1/R^2 loss" - that is not true in most real-world listening situations. For example, in a room, almost all the sound that you hear is reflected from the walls, floor and roof, not coming direct from the instrument. Even out of doors, you get reflections from the ground and from buildings, trees, etc, and refraction of sound energy radiated "upwards" into the air, because air temperature and wind speed both vary with the height above the ground.
– user19146
Jul 6, 2018 at 13:38
• Jul 7, 2018 at 15:02