An Introduction To The Decibel

The unit most commonly used (and often misused, or at least misunderstood) to compare sound levels is the decibel (abbreviation dB).

Most confusion arises from the fact that - on its own - the decibel is not a unit of measurement at all. It is a mathematical tool for comparison (like percentage). The question: "How many percent is the output of this amplifier?" makes no sense: how many percent of what? On its own, the question "How many decibels is the output of this amplifier?" makes no sense either. To make the question sensible, we need to ask in terms of comparison: "How many decibels is it greater (or less) than something else?". To understand the answer, we also need to know what a number expressed in decibels actually tells us.

While we have tried to cover the basics below, a more detailed explanation is beyond the scope of this section. For a fuller description (including more worked examples), try chapter 3 of The Sound Reinforcement Handbook (Yamaha), by Gary Davis & Ralph Jones.

What is a decibel?

A decibel is one tenth of a Bel (a Bel - named after Alexander Graham Bell - is the logarithm of an electric or acoustic power ratio).

While this may sound complicated (the decibel is a fraction of the logarithm of a ratio), the relationship between the numbers is always the same (like the relationship between Celsius and Fahrenheit), and - like temperature - the real-world events that give rise to the numbers can usually be seen and felt, as well as measured and compared. To work out the numbers for yourself, all you need is a calculator with:

• Logarithms (log)

• Powers (this function is usually denoted by a "^" sign).

Decibel calculation

Decibel values can be calculated from any power measurements that use a common linear scale (e.g. Watts). In all cases they are derived from the ratio between two measurements, and in all cases they are calculated by finding the logarithm of the ratio, and multiplying it by ten*.

The decibel can also be used to compare pressure measurements (e.g. Volts, or sound pressure), and is calculated in the same way, save that the logarithm is instead multiplied by twenty. This reflects the fact that any power value corresponds to the square of a pressure value: watts are calculated by squaring the voltage and dividing the result by the resistance (P = V^2/R, where P is the power in Watts, V is the voltage in Volts, and R is the resistance in Ohms)**.

*In case you wondered why you are multiplying by ten to find a tenth of a Bel, note that a millimetre is one tenth of a centimetre: you would multiply a measurement given in centimetres by ten to obtain the value in millimetres.

**See the worked example below.

Graphic llustration of decibel calculation

To calculate the difference in decibels:

For power (e.g. watts)

• Divide the larger power by the smaller power.

• Find the logarithm (base 10) of the result.

• Multiply the logarithm by 10.

For pressure (e.g. volts, sound pressure)

• Divide the larger pressure by the smaller pressure.

• Find the logarithm (base 10) of the result.

• Multiply the logarithm by 20.

If you need to work the other way (i.e. to work out what value is n decibels larger or smaller than a reference value):

For power (e.g. watts)

• Multiply your reference value by 10^(n/10).

For example, if your system is rated at 100 watts, a 20 decibel increase represents 100 x 10^(20/10) watts = 100 x 10^2 watts = 100 x 100 watts = 10,000 watts!

Negative values work in exactly the same way, so that if your system is rated at 100 watts, a reduction of 20 decibels (−20 dB) will produce 100 x 10^−(20/10) watts = 100 x 10^−2 watts = 100 x 0.01 watts = 1 watt. At 20 decibels below full output, your 100 watt system is only running at 1 watt!

For pressure (e.g. volts, sound pressure)

• Multiply your reference value by 10^(n/20).

For example, if your microphone produces 2.6 millivolts, an increase of 60 decibels will produce 2.6 x 10^(60/20) millivolts = 2.6 x 10^3 millivolts = 2.6 x 1,000 millivolts = 2,600 millivolts = 2.6 volts.

Where power and pressure combine, the values remain consistent.

For example, an amplifier produces an output of 20 Volts. From this, we can calculate the power generated into a 4Ω load as (20^2)/4 Watts = 400/4 Watts = 100 Watts.

If we double the output voltage (40 Volts) we find that the power generated is now (40^2)/4 Watts = 1,600/4 watts = 400 Watts.

Using decibels we find that doubling the output voltage is an increase of 6dB (20 x log 2 = 20 x 0.3 = 6). However, we have quadrupled the power generated. Yet using decibels we find that the increase is still the same: 6dB (10 x log 4 = 10 x 0.6 = 6).

Fixed decibel values

Sometimes the decibel is used to compare measured values to a single fixed reference value. Where this is the case, a suffix is usually used to indicate the value that is being used as a reference (e.g. dBm, dBV, etc.). Common suffixes in the audio industry are:

dBm	Reference value (0dB) = 1 milliwatt. For example, 10 dBm = 1 x 10^(10/10) milliwatts = 1 x 10^1 milliwatts = 1 x 10 milliwatts = 10 milliwatts.
dB SPL	Reference value (0dB) = 0.0002 dynes/cm². A Sound Pressure Level (SPL) of 0.0002 dynes/cm² is the threshold of hearing: the smallest perceptible sound. It is the level at which someone with perfect hearing can first detect a 1kHz tone. You may also see dB L*_p, which is an alternative abbreviation for Sound Pressure Level.
dBu	Reference value (0dB) = 0.775 volts (775 millivolts). The "u" stands for "unloaded", and today is used almost universally in preference to dBv, from which it derives (see below). For example, +4 dBu = 0.775 x 10^(4/20) volts = 0.775 x 10^0.2 volts = 0.775 x 1.58 volts = 1.23 volts.
dBv	Reference value (0dB) = 0.775 volts (775 millivolts). Historically, a common impedance in professional audio was 600Ω. 775 millivolts across a 600Ω load produces 1 milliwatt, so in a 600Ω circuit, 0dBv = 0dBm. However, the distinction between lower- and upper-case letters ( dBv and dBV) can easily be missed, so it is safer to use dBu instead.
dBV	Reference value (0dB) = 1 volt. For example, −10 dBV = 1 x 10^−(10/20) volts = 1 x 10^−0.5 volts = 1 x 0.316 volts = 0.316 volts.

*One dyne is the amount of force required to accelerate a mass of one gram at the rate of one centimetre per second per second. The threshold of hearing - 0dB (SPL) - is 2^-4 (2 ÷ 10,000) dynes per square centimetre, which in terms of everyday pressures you can feel is a tiny fraction of Not Very Much. For this reason, dynes/cm² is generally used to describe sound pressure, as expressed using other units the number is Even Smaller. If you have been raised on other units, 0dB (SPL) = 2 x 10^-4 dynes/cm² = 2 x 10^-4 μbar (microbar) = 2 x 10^-5 N/m² (Newtons per square metre) = 2 x 10^-5 Pa (Pascals) = 20 μPa (micropascals) = 2.9 x 10^-9 psi (pounds per square inch, if you really need to know).

Technically, when using the decibel to make comparisons, the original unit of measurement should always be given to make sense of the comparison: if someone tells you that one banana is 10 more than another banana, do they mean ten times the weight, or 10 times the price (or just 10g heavier)? In practice, however, the unit of measurement is often left out in decibel values, and in sound systems this is no great disaster. If bananas are sold by weight, ten times the weight is ten times the price, and vice versa. In sound systems, similarly, differences in one part of the system generally translate directly to other parts of the system. Without compression or expansion, a 3dB difference in the volume of an instrument will produce a 3dB difference at the microphone, a 3dB difference at the mixer channel, a 3dB difference at the amplifier, and a 3dB difference at the listener's ears, even though these may all use different units of measurement.

However, the decibel is sometimes used to describe sound pressure levels without stating the unit of measurement or the point of reference: "the sound reached 115 decibels in the fourth row", or "the output of the Fictional combo is 115dB".

More often than not, this means the sound is either:

115dB SPL (sound pressure measured using an unweighted scale, reference value 0.0002 dynes/cm²).

115dBA (sound pressure measured using an A-weighted scale, reference value 0.0002 dynes/cm²). An A-weighted scale is one which compensates for the fact that the human ear does not hear all frequencies equally well, so that sounds at the lowest and highest audible frequencies have a lower value than the same sounds using an unweighted scale.

However, three decibels louder would be three decibels louder whatever unit or reference point was used.

Why use decibels?

The most common questions the decibel is used to address are "How loud is it?", or "How much louder is it?".

The same question could be answered in a number of ways: it is MUCH louder, as loud as a jet engine, nearly twice as loud as Glastonbury last year. However, these answers are subjective and approximate. Not all jet engines are equally loud, they all get louder as you get closer (how far away was the listener?), and not everyone has heard one. Similarly, not all people experience "twice as loud" in the same way (to some it is a bigger difference than to others), and not everybody was at Glastonbury last year.

The decibel is useful because it gives us a number - like inches or metres - that is replicable (the same event always yields the same answer to anyone using the same measure). However - unlike inches or metres - the decibel does not have a single fixed reference point. While some common fixed reference points exist (for example dBA, dBm, dBV, dBu: see above), decibels can also be used to compare levels in any system, with replicable results. The illustration below shows how decibel values (in this case dBV) correspond with a linear scale (in this case Volts):

Note that at the bottom end of the scale a 6dB difference represents a change of one volt, while at the top end a 6dB difference represents a change of sixteen volts. If we redraw the scale so that decibels rather than volts are of equal size, it looks like this:

The decibel is convenient for comparing levels in sound systems because the greatest sound pressure the ear can accommodate without discomfort is around 1,000,000 (yes, a million*) times greater than the smallest sound pressure the ear can detect. However, we can't distinguish a million different levels (or even a few hundred different levels) over this range. Expressed in decibels, the smallest sound pressure we can hear (our reference point) is 0 dB**, and sound pressure that is uncomfortably loud - a million times greater than our reference point - is 120 dB. We can just about distinguish 120 different levels over that range: our hearing doesn't easily distinguish between differences in level of less than about 3 decibels, but with practice differences of 1 decibel (about 9/8 difference in sound pressure) are perceptible. To most listeners a difference of 1 decibel is "just noticeable", 3 decibels is "clearly noticeable", and 10 decibels is "twice as loud".

Also, we experience sound in a way that corresponds more closely with a logarithmic scale than with a linear scale: if we listen to ten successive doublings of sound pressure, they sound like ten equal increases. Viewed on a linear scale, the tenth increase would be 512 times bigger than the first, but it doesn't sound 512 times bigger: it sounds the same. It makes sense, therefore, to use a logarithmic scale when we compare sound levels: the numbers match our experience. In power terms, 1 decibel is roughly the difference between 10 Watts & 12.5 Watts, and this difference in volume sounds about the same as the difference between 1,000 Watts & 1,250 Watts (also 1 decibel).

*Some authorities give the range as 130 dB (approximately 3 million times), and a few as 140 dB (10 million times). If you reckon you can stand 130 dB (SPL) at 3.15 kHz, I'll take your word for it. Don't try to prove it to anyone, even yourself. It will permanently damage your hearing.

**If a measured value is the same as our reference point then (measured value) ÷ (reference point) = 1. The logarithm of 1 is 0 (1 = 10^0). Anything multiplied by 0 (10 or 20) equals 0. So our reference point, being the same as itself, always has a value of 0 dB. 0 dB can be thought of as "no difference", or "no change" (but it is NOT "nothing" or "no value").

How loud is a decibel?

As a rough guide, the following levels (SPL) approximately correspond with the sounds described:

Value	Difference from reference level	Description
0dB SPL	1 x reference (unity)	The threshold of hearing (equivalent to a pressure of 0.0002 dynes/cm2). This is the quietest sound that a child or young adult with good hearing can detect at 1kHz. It is not silence (although in terms of how loud it is in a P.A. system it might as well be): it is the level from which all other values using the decibel (SPL) scale are calculated.
10 - 20dB SPL	3 - 10 x reference	A gentle breeze through the trees.
20 - 30dB SPL	10 - 32 x reference	A soft whisper (at 1 metre).
30 - 40dB SPL	32 - 100 x reference	A quiet auditorium.
40 - 60db SPL	100 - 1,000 x reference	Background music in a cafe, bar or restaurant.
60 - 70dB SPL	1,000 - 3,200 x reference	Typical conversation levels (from the listener's position).
70 - 80dB SPL	3,200 - 10,000 x reference	The cabin of an aircraft during normal cruise conditions.
80 - 90dB SPL	10,000 - 32,000 x reference	Typical wedding or dinner-dance band (typical audience position).
90 - 100dB SPL	32,000 - 100,000 x reference	Loud orchestra (playing fff, as it would sound in the front row of the audience).
100 - 110dB SPL	100,000 - 320,000 x reference	Typical disco.
110 - 115dB SPL	320,000 - 560,000 x reference	A loud rock band (front rows of audience).
115 - 130dB SPL	560,000 - 3,200,000 x reference	Threshold of pain. Often given as 120 dB SPL, this varies with frequency, and from person to person.
140dB SPL	10,000,000 x reference	Jet engine from 3 metres.

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