Way back at the beginning of the electronics age (even before I was born) engineers needed a way to easily express the levels of signal losses and gains in all the new fangled thingamabobs that they were inventing like telephone lines, radio transmitters, antennas, band pass filters, and electric bass amplifiers, just to name a few. The problem is that some of these components (like for example an electric bass amplifier) took a very small signal and amplified it into a very large signal, usually in several stages with each stage having its own set gain.
If these gains were expressed in simple ratios or as a multiplier, they could end up being very big numbers (or very small numbers in the case of negative gain/attenuation) because the final overall ratio would be the product of all the stages multiplied together (i.e., stage 1 times stage 2 times stage 3, etc.). This could get very complicated depending on the number of stages and the ratios involved.
A simpler approach is to take the logarithm of each ratio and add them together (e.g., log [stage 1] + log [stage 2] + log [Stage 3], etc.). This works because when you multiply numbers together you are actually adding their logarithms. The logarithm of the ratio of two power levels is the definition of the bel. The bel was defined by AT&T (and named after Alexander Graham Bell) as a way to express signal losses over miles of cable in telephone systems. As it turns out, it is more convenient to use tenths of a bel (B), or decibels (dB). The formula for decibels is:
dB = 10 log (P1/P2)
In this formula P1 and P2 are the levels of power (expressed in a common unit such as watts) that are being compared. As a practical matter, you will probably never have to use this formula unless you build your own amplifiers. Here are some easy to remember conversions that might come in handy as applied to power ratios:
0 dB = unity (no gain and no attenuation)
+3 dB = double the power (x2)
-3 dB = half the power (x 0.5)
+6 dB = four times the power (x 4)
-6 dB = one fourth the power (x 0.25)
+9 dB = eight times the power (x 8)
-9 dB = one eighth the power (x .125)
+10 dB = ten times the power (x10)
-10 dB = one tenth the power (x0.1)
+20 dB = hundred times the power (x100)
-20 dB = one hundredth the power (x0.01)
Let's say that you have a parametric equalizer on your bass amp that has a knob with a scale that ranges from -12 dB to +12 dB with a mid point of 0 dB. If the knob is set at the midpoint of 0 dB, then the level of the output of the of the affected frequency will be equal to the level of the input (what ever it is) for the set frequency (that‘s another knob). Setting the knob at +3 dB or -3 dB will double or halve (respectively) the input power for the affected frequency. Each 3 dB increment will double or halve the signal again resulting in +12 dB being sixteen times the input and -12 dB being one sixteenth (x 0.0625) the input.
So far, we have only discussed decibels (dB) as a relative number used to express the ratio between two quantities. It can also be used to express the ratio of a measured quantity to a reference quantity. To do this, a qualifier is added to the dB symbol. One of the ones that you might come across pertaining to bass equipment, or some of your other audio electronics, is dBSPL where "SPL" stands for Sound Pressure Level. This is commonly used to specify the strength of the sound waves being produced by a speaker.
The reference level (0 dBSPL) is the lowest level of sound that is audible to the human ear. It has been described as the sound that a mosquito makes at three meters. The sound level at 120 dBSPL is loud enough that it starts to cause pain. So, lets say that the comfortable range of loudness for human hearing is between 0 dBSPL (barely audible) and 120 dBSPL (threshold of pain). This means that the upper level (120 dBSPL) is 1,000,000,000,000 (one trillion) times louder than the lower level (0 dBSPL). That's the sound of one trillion mosquitoes three meters from your head!
In my mind, that's a lot worse than Hendrix's humming bird! It is important to note that Sound Pressure Level and apparent loudness are not the same. Human hearing is logarithmic in nature. A 1dB change in SPL is not discernable by most people, a 2 dB change in SPL is discernable by most people, and a 3 dB change in SPL is easily discernable. Doubling (+3 dB) the SPL does not double the apparent loudness of a sound. It takes a ten fold increase (+10 dB) in SPL to double the apparent loudness of a sound.
For more information on decibels see:
http://en.wikipedia.org/wiki/Decibel
For more information on Sound Presure Level see: http://en.wikipedia.org/wiki/Sound_pressure
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