One purpose might be pitch detection.
When several frequencies are combined that are small integer multiples of a fundamental frequency, we perceive the combination as having pitch given by that fundamental.
For something like a guitar string, the frequency components are close to small integer multiples of a fundamental frequency.
But for something like a church bell, the frequencies don’t line up so neatly, though there’s still a clearly perceived pitch.
For something like a metal mixing bowl, it may be difficulty to predict what pitch a person will hear when something strikes the bowl.
One complication we haven’t addressed yet is that the fundamental frequency will not be unique without some constraint.
In the example above, the frequencies were all multiples of 440, but they’re also all multiples of 440/n for every positive integer n.
We might get around this by specifying some lower bound on the fundamental frequency.
Or we could say that all other things being equal, we want the largest candidate for the fundamental frequency.
We could formulate the problem of finding the fundamental frequency as an optimization problem.
For example, we could form a mixed integer program.
Suppose we have three frequencies f1, f2, and f3.
We could find a fundamental frequency f and integers n1, n2, and n3 that minimize (f1 – n1 f)² + (f2 – n2 f)² + (f3 – n3 f)² subject to a lower bound on f.
This minimizes the squared distances to multiples of a fundamental frequency, but maybe we want to emphasize relative distance rather than absolute distance.
If we denote the nearest integer to x by [x], then we might want to minimize (f1/f – [f1/f])² + (f2/f – [f2/f])² + (f3/f – [f3/f])² .
The first formulation has a more common form.
The latter has a more complicated objective function, but it’s only a function of one variable.
Here’s what the latter looks like for frequencies 1760, 2200, and 3080.
Clearly there’s a minimum at 440 Hz.
Here’s the same plot with 10% random noise [1] added to each frequency: 1701, 2368, and 3339.
Now there’s a minimum near 336, but the local minimum at 566 is nearly as good.
Related posts Tonal prominence in a leaf blower Optimization, dominoes, and Frankenstein [1] There are a couple reasons you might want to solve a problem like this.
Maybe your frequencies really are integer multiples of a fundamental frequency, but there is measurement error.
Another is that the frequencies are not exactly multiples of a fundamental, as when striking a bell or a mixing bowl.
How might you formulate the two cases differently?.