For the majority of humans, there is nothing simpler than listening to sound. You simply, well, listen.
When it becomes necessary to describe the listening experience analytically, however, a host of complex equations and diagrams are required to describe even the simplest of sonic events.
The benefit of mathematical analysis is that it can yield insights that are not apparent through intuition alone. Acoustic signals are easily measured, and the audio components that produce them have characteristics that can be measured.
We do not expect specifications to tell us how a product sounds. This is what listening is for. The main purpose of specifications is to allow us to make sure that we have the right tool for the job, and this information is most often presented in the form of charts and graphs.
But what does this information really mean?
The heart of understanding the specification sheets that describe audio products is the understanding of dependent and independent variables.
The concept is one that most people use every day, though often without realization.
An independent variable is one that describes a series that has a fixed value. For example, the time of day in the city that you live in is an independent variable. Regardless of what happens tomorrow, time will progress like it did today.
What will change are your moment-to-moment activities. These events represent a dependent variable. They depend on time.
If you look at a page in your day planner, you are looking at a plot of activities vs. time.
Time is the independent variable. It is the same on every page of the planner. The scheduled events are the dependent variables, because where you go and what you do depends on what time it is. Most graphs show the relationship between dependent and independent variables.
Now let’s look at a variation on the theme. Let time be the independent variable (it usually is) and let the loudness of the sound system during a show be the dependent variable.
The plot might look something like Figure 1.
Figure 1: In this example, time is the independent variable while loudness is the dependent variable (click to enlarge)
The horizontal axis represents time (the independent variable) and the vertical axis represents loudness (the dependent variable).
We will call the horizontal axis the x-axis and the vertical axis the y-axis, although any two letters would do. The values on each axis are usually discrete, meaning that they are individual samples, points, or measured values called data points.
The fact that most graphs look like squiggly lines just means that after many data points were taken, they were joined with a line to make it easier to read.
Such two-dimensional plots are found on virtually every good specification sheet in existence. They simply answer the question “What is the value of y when the value of x is this?” Some examples of two-dimensional plots found in audio engineering include:
Each plot shows the value of y for a given value of x. Pretty cool. In math-speak, in each case it can be said that y is a function of x. (We sound smarter when we say it like this.)
From this example, it can be seen that frequency is a very common independent variable in the world of audio and acoustics. The y parameters are said to be frequency-dependent.