Introduction to Computer Music: Volume One

Decibels:

While power is measured in watts, the most-used acoustic measurement for intensity is the decibel (dB). Named in honor of Alexander Graham Bell, a decibel = 1/10 of a bel. A decibel is a logarithmic measurement that reflects the tremendous range of sound intensity our ears can perceive and closely correlates to the physiology of our ears and our perception of loudness. There are many different forms of decibel measurement and it is not always clear which method of computation is being used unless it is labeled properly.

I must admit that I was once intimidated by logarithms, but with cheap calculators to do the math (one previously used log tables), just a simple understanding of how they work is all that is necessary for decibel calculations.

A logarithm primer

can be thought of as "what power of 10 will result in x." For example, because 102 = 100. Decibels are often used to measure very minute values, which can also be expressed by logs of negative numbers. For example, and , a value we will use for our threshold of hearing measurement below. If it is expressed then .

A decibel is a measurement used to compare the ratio of intensities of two acoustic sounds (or electronic signal). The ratio (R) of two signals expressed by their power in watts (W1 and W2) is:

There are many different types of decibel measurements, so for the purpose of clarity, the above form, which measures power or intensity is called dBm. For the purpose of having a standardized absolute measurement of power (i.e. a comparison not to another signal, but to an industry-fixed value), the nominal reference wattage (W2)has been defined as 1 miliwatt (0.001 watt). In absolute terms, a 1-watt signal, which has 1,000 times the power of the reference wattage, will be 30 dB, computed below:

dBm=10 log10 (1 watt/.001 watt) dBm =10 log10 (1000) dBm=10 x 3 [because log10 1000 = 3] dBm=30

dBm is the form most commonly used to evaluate power in audio circuits.

Since intensity (I) at a fixed distance of measurement is directly proportional to power, a similar measurement can be made:

In this case, a doubling of power equals an increase of +3dB. When we study filters later on, you will notice that a filter cut-off frequency is defined as the half-power point, which is calculated as –3dB.

While the original dB scale was created for comparison of intensity or power, it is also commonly used as a measurement of amplitude (A) or sound pressure as defined above. The formula for computing relative amplitude or sound pressure is:

By comparing this formula to the one for dB above, the relationship between amplitude, power and intensity becomes clear. In this case, a doubling of amplitude from one source to another equals an increase of +6 dB as shown below:

The most common acoustic ratio measures a current sound against a predetermined value of the threshold of audibility mentioned above but expressed as 2 x 10^-12 watts. This absolute measurement is referred to as the sound-pressure Level (SPL) and gives us a means of generalizing relative loudness of common acoustic sources (note that the dB is followed by SPL to indicate this mode of measurement). The logarithmic scale from the threshold of hearing to the threshold of pain ranges from 0.00002 N/m^2 to 200 N/m^2, or about 120-130 dB SPL, at which point the entire body, not just the ears sense the vibrations (NB: In preparing this article, it quickly became apparent that no “standard” for the threshold of feeling or the threshold of pain has been established, and in fact ranges in the references used from 120 dB SPL to 140 dB SPL, which is a huge variation of opinions and points out the differences between acoustic and psychoacoustic measurement). Younger people also have more effective protection mechanisms and so can tolerate louder sounds (surprise!).

If we accept 130 dB as the threshold of pain, then humans hear sounds that range from the smallest perceptible amplitude to those that are 10,000,000,000,000 as loud or 10 watts/m². Both the dB and dB SPLscales reflect the incredible discrimination of human hearing, our most sensitive sense by far.

Here are some vague benchmarks (which of course depend on many factors, including the listener’s distance from the sound).

Source	Power (watts/m2)	dB SPL
Threshold of pain	10	130
Jet takeoff from 500 ft.	1	120
Medium-loud rock concert	.1	110
Circular saw	.01	100
New York subway	.001	90
Jack-hammer from 50 ft.	.0001	80
Vacuum cleaner from 10 ft.	.00001	70
Normal conversation	.000001	60
Light traffic from 100 ft.	.0000001	50
Soft conversation	.00000001	40
Whisper from 5 ft.	.000000001	30
Average household silence	.0000000001	20
Breathing	.00000000001	10
Threshold of hearing in young	.000000000001	0

Amazing Factoid: The Ben and Jerry's Ice Cream Company recently funded reserach on a a freon-free freezer which uses sound waves pumped in at an astonishing 190 decibels to compress the air enough to bring the temperature down to 0 degrees.

Signals from microphones, most of which seek to accurately transform changes in SPL to proportional changes in voltage (V), can also be measured by the same method. If one were to change the miking distance to the sound source, the voltage differences could be measured as follows:

If measured properly, halving the distance of the mic to the source, thanks to the inverse square law should double the voltage produced by the microphone, giving a +6 dB increase in amplitude (which if you’ve been reading closely also produces four times the intensity). For a standardized comparison of voltages, 0.775 volts is used as the reference level for = 0 dB.

We have looked at two basic types of dB measurement, one for power and intensity, and the other for amplitude, SPL and voltage. Several other weighted dB scales, such as dBA are used for specific purposes, such as more closely mirroring the way we hear, but this will be discussed in further detail in the psychoacoustics sections.