And how are they different from linear scales?
Have you ever wondered why we use logarithmic scales to measure certain quantities? Well, let’s dive into this fascinating topic and uncover the reasons behind it.
First, let’s take a look at the difference between linear and logarithmic scales. A linear scale is what we encounter in our day-to-day lives. It follows a constant rate of increase, where each unit on the scale represents an equal increment. For example, if you have a ruler with centimetre markings, each centimetre represents the same distance.
On the other hand, a logarithmic scale expands at an increasing rate as it moves out from the starting point. In a logarithmic scale, each unit represents a multiplication (or division) by a constant factor, rather than a fixed amount. In other words, the intervals on the scale increase exponentially.
A good way to visualise this difference is to look at geometric (linear) and logarithmic spirals.
The Geometric or Arithmetic Spiral
A geometric spiral is a spiral that expands at a constant rate, like a linear scale. A geometric spiral is also known as an Archimedean Spiral, or an arithmetic spiral.
The points on the spiral move away from the centre by the same amount with each complete loop, like this:
This pattern occurs in a roll of paper or tape of constant thickness wrapped around a cylinder.
The coils of watch balance springs and the grooves of very early gramophone records form Archimedean spirals, making the grooves evenly spaced (although variable track spacing was later introduced to maximize the amount of music that could be cut onto a record).
The Logarithmic Spiral
A logarithmic spiral is a spiral that expands at an increasing rate as it moves outward from the centre. In fact, they increase in a geometric progression, which means each loop is further from the centre than the previous one by a multiple of the previous distance.
A logarithmic spiral, or growth spiral looks like this:
The logarithmic spiral is also called the spira mirabilis, Latin for “miraculous spiral”, named that for the unique property that while the size increases with each loop, the shape of each successive loop or curve is the same. One outcome of this property is that the resolution at the lower end of the signal is expanded, and you can see into the lower-level signals more clearly than in a linear scale. In other words, the logarithmic scale allows the representation of a wide range of values in a compact and intuitive way. Nature is filled with examples of logarithmic spirals, such as nautilus shells, sunflower heads, and the arms of spiral galaxies like the Milky Way.
Given the prevalence of the logarithmic spiral in nature, it’s not surprising that a number of things we measure are best represented on a logarithmic scale.
In scientific and engineering fields logarithmic scales offer distinct advantages. For instance, the Richter scale used to measure the intensity of earthquakes is a logarithmic scale. Each increase of one unit on the Richter scale represents a ten-fold increase in the magnitude of the earthquake. Therefore, an earthquake with a magnitude of 6 is 10 times more powerful than one with a magnitude of 5, and 100 times more powerful than one with a magnitude of 4.
Another example of the use of logarithmic scales can be found in the field of acoustics. The decibel scale, which is used to measure sound intensity, is a logarithmic scale. The decibel scale allows us to represent the wide range of sound levels that the human ear can hear, from the faintest whisper to the loudest rock concert, in a compact and intuitive way.
Some other uses for logarithmic scales:
- In photography for counting f-stops for ratios of photographic exposure
- In electronics using the Neper measurement for comparing voltages, currents and power levels
- In chemistry to measure acidity as pH
- In astronomy to measure brightness of stars using the stellar magnitude scale
Texcel measures Sound Pressure Level in dBL or Linear decibels
Isn’t that now a contradiction in terms? We just said that the decibel scale is logarithmic and now we talk about linear decibels.
The answer is that the Linear in dBL defines the frequencies we measure, not the units we measure them in. Because they are sound waves we measure them in the dB logarithmic scale – hence the dB. Because we do not filter the low frequencies we call the measurements linear – hence the L.
What do we mean by the statement that we do not filter the low frequencies?
All pressure waves in air are related to sound, and sound is usually considered to be ‘what the human ear can hear’. Like our eyes and ‘light waves’, our ears have evolved to hear most clearly the range of frequencies that are most useful for us to hear. These would be largely those frequencies that other humans can produce with their voice box, so we can hear what they say. There would also be a need to hear approaching danger, like the rustle of a snake in the grass, the sound our predators make to communicate and move around.
So, to best measure sound waves that annoy humans, sound measuring devices filter the frequencies of sound to best match the human ear. There are a few filters used in Sound Level Meters, A-Weight being the most common. These act to cut out the low frequencies and high frequencies the human ear can’t hear.
But the Sound Pressure levels the Texcel monitors record are not concerned with the human ear. The concern here is the potential for the Sound Pressure levels to cause damage to structures. While sound waves that can damage structures are produced in nature, thunder for example, the real structural damage is carried in sound waves that are of too low a frequency for us to hear, often called infrasound. In other words, we want all frequency levels to be reported equally, not filtered to match the human ear. This type of measurement is called a linear dB measurement.
Having said that, we do filter the very high frequencies, because we know that they do not carry enough energy to damage our structures. And we don’t measure below 2 Hz in accordance with the Australian and international standards.
As Galileo said, ‘Measure what can be measured.”
Then he added: “and make measurable what cannot be measured”, but that takes a while …