The Decibel

An Introduction to the Decibel and its Use in Live Sound Reinforcement

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.

What is a Decibel?

A decibel is one tenth of a Bel. The Bel (named after Alexander Graham Bell, who used a logarithmic scale to quantify power losses in long cables) 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 Calculations

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:

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.

To calculate the difference in decibels between two values:

For power (e.g. watts)

• Divide the second value by the first value^‡.
• Find the logarithm (base 10) of the result.
• Multiply the logarithm by 10.

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

• Divide the second value by the first value.
• Find the logarithm (base 10) of the result.
• Multiply the logarithm by 20.

‡ The first value is your reference value. This may be a measured value, or may be a common standard reference point (e.g. dBu, see below).

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 × 10^(20/10) watts = 100 × 10^2 watts = 100 × 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 × 10^−(20/10) watts = 100 × 10^−2 watts = 100 × 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 × 10^(60/20) millivolts = 2.6 × 10^3 millivolts = 2.6 × 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 × log 2 = 20 × 0.3 = 6). However, we have quadrupled the power generated. Yet using decibels we find that the increase is still the same: 6dB (10 × log 4 = 10 × 0.6 = 6).

You can find functions to carry out most of the above calculations without needing the formulae or a calculator on our System Calculations page.

Fixed Decibel Values

Sometimes the decibel is used to compare measured values to a single fixed reference value (or to state values with respect to that 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:

Technically, when using the decibel to make comparisons, the unit of measurement should always be stated 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, a 3dB difference at the loudspeaker, and a 3dB difference at the listener's ears, even though these may all have different reference points and/or 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 2 × 10″ Combo is 115dB at 1 metre’.

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).

or

115dBA (sound pressure measured using an A-weighted scale, reference value 0.0002 dynes/cm² @ 1 kHz). 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 any 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 and reference point). 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 decibel is also more useful than linear measures for comparing levels in sound systems for two important reasons:

1. Our hearing can accommodate a very wide range of levels. 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’.
2. We experience sound in a way that corresponds more closely with a logarithmic 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 140 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’).

The illustrations below compare linear scales (Volts, in the first example) with decibels:

As discussed above and elsewhere, power is calculated from the square of the voltage, so when we look at Watts the range of values is even greater:

We can also look at the relationship the other way round (where the values on our linear scale are equally spaced):

Here, we can see that at the bottom end of the scale a change of one volt accounts for a 6dB difference, while at the top end a change of twenty volts only accounts for a 4dB difference.

The Decibel in Use

Before you spend much time using professional sound equipment, you will find the decibel almost everywhere you look: on gain and EQ controls, faders, meters, amplifiers, and digital menus in everything that has them.

Having an understanding of decibels will therefore make using audio equipment very much easier. While some aptitude in mathematics will obviously be an advantage, just having a working knowledge of a few everyday relationships will help, so, if you can, at least try to memorise some of the following common values:

 

*The relationship between decibels and hearing is approximate. Not all listeners will describe a 10dB difference as "twice as loud", or select a 10dB difference as the point at which a sound has doubled in volume.

The following sound samples illustrate these values:

1. Pink noise with stepped level changes.

   This is a recording of pink noise at a background reference-level, with three-second bursts at higher levels. Each increase is preceded by an announcement of the level difference in decibels from the background level. Because the tone and dynamics of the sound are constant and level changes are abrupt (and announced), even a 1dB difference is fairly readily noticeable. Where the level changes gradually, the difference is less noticeable, as in this sample varying by 1dB:

2. Music sample played at different levels.

   Here, an initial reference (0dB) music sample is played back at increasing levels separated by short periods of silence. Where there is greater dynamic variation than that provided by a tone or pink noise signal, a 1dB difference would not be readily noticed by most listeners (even in this example, where the samples are otherwise identical). In a live music event, increasing the level by 1dB on the main faders between one song and the next would almost certainly go unnoticed. Note that the stated levels are all referred to the initial sample:

How Loud is a Decibel?

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

 

See Also...

You can find a more detailed explanation of the decibel (including more worked examples) in chapter 3 - ‘The Decibel, Sound Level, and Related Items’ - of The Sound Reinforcement Handbook (Yamaha), by Gary Davis & Ralph Jones.

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