# What is "frequency"?

Pattern-chaser
Setting aside pure sine waves, and looking instead at real-world sound, such as music, I wonder what "frequency" is? Fourier's Theorem seems to be aimed specifically at PERIODIC waveforms, but music (as just one example of real-world sound) is not periodic, as far as I can see. So it is not clear to me that Fourier's Theorem applies to music, even though we have achieved many working applications using it.

So, in the context of music, what is "frequency"?

davenn

Mentor
Music is not periodic? I can't imagine what would make you think that.

nasu and Pattern-chaser
Pattern-chaser
Music is not periodic? I can't imagine what would make you think that.
Looking at a music waveform on a 'scope offers a simple and real-world illustration. "Periodic", as applied to waveforms, means that the waveform repeats cyclically, and music does not do this. ... Unless you consider an entire symphony (for example) as a single cycle of a waveform that could (theoretically) repeat over and over, except that it doesn't.

davenn and PeroK
Mentor
Looking at a music waveform on a 'scope offers a simple and real-world illustration.
You wouldn't be able to view it on the scope if it wasn't periodic.
"Periodic", as applied to waveforms, means that the waveform repeats cyclically, and music does not do this. ... Unless you consider an entire symphony (for example) as a single cycle of a waveform that could (theoretically) repeat over and over, except that it doesn't.
Of course music repeats cyclically! Middle C is 264 Hz, so a quarter note at 120 beats per min is 132 cycles.

Pattern-chaser
Pattern-chaser
You wouldn't be able to view it on the scope if it wasn't periodic

Of course music repeats cyclical! Middle C is 264 Hz, so a quarter note at 120 beats per min is 132 cycles.
But you *can't* view it on a 'scope unless you do a single-scan. On continuous scan, the scope can't sync, possibly because the music waveform isn't periodic?

Also, middle C is indeed 264 Hz, but no instrument (excluding electronic ones) produces a 264 Hz sine-wave when middle C is played on it. Instead, we see a complex waveform which does not appear to be periodic. [And this complexity is not just 'harmonics', but also includes all sorts of instrument-dependent components, and maybe ambient noise, etc, too.]

davenn
Gold Member
Looking at a music waveform on a 'scope offers a simple and real-world illustration. "Periodic", as applied to waveforms, means that the waveform repeats cyclically, and music does not do this
Music is a (mixed) sum of various waveforms (some longer, some shorter) with different frequencies and different timbre. In the previous description I exclude some effects which complicates matters further, like reverbs and delays, which adds modified time delayed copies of sounds to the original sounds.

Edit:

And that's why the final waveform of a musical piece looks quite chaotic when you look at it. But of course there are frequencies present in the waveform, they're just heavily mixed together. For instance, you can analyze the piece with a spectrum analyzer (FFT) to see which frequencies are more prominent than others.

Also, there is almost always some kind of tempo involved, and that is also a sort of a overall, general frequency of the piece, so to say.

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PeroK and Pattern-chaser
Pattern-chaser
But of course there are frequencies present in the waveform, they're just heavily mixed together. For instance, you can analyze the piece with a spectrum analyzer (FFT) to see which frequencies are more prominent than others.
Ah, but can you? Fourier's Theorem applies only to periodic waveforms, which brings us back to my original question. In the context of non-periodic real-world sound, e.g. music, what is "frequency"?

Gold Member
Ah, but can you?
Yes, a BIG yes. I've done it hundreds of times (probably well over a thousand).
It's essential when producing music (e.g. in the mastering process).
Here's one spectrum analyzer, for instance.

Edit: You can also do it in the free software Audacity. Try it, you might like it.
The function is called "Plot Spectrum" in Audacity.

Pattern-chaser
Pattern-chaser
Yes, a BIG yes. I've done it hundreds of times (probably well over a thousand).
It's essential when producing music (e.g. in the mastering process).
Here's one spectrum analyzer, for instance.

Edit: You can also do it in the free software Audacity. Try it, you might like it.
My fault. I should've written "Ah, but is it VALID to apply Fourier's Theorem, when music is non-periodic, and that theorem applies only to periodic waveforms?" I'm not saying it doesn't work; it does. But is it valid physics?

Edit: I've used Audacity a few times - a very handy utility!

DennisN
Gold Member
You can extend the "simple" definition of frequency from Fourier analysis to non-periodic signals.
One way of doing it is to look at the instantaneous frequency (the derivative of the phase); another way of doing it is to simply divide the time series into "chunks" of data before calculating the spectra; this it often how DSP is done.
Time series analysis (and estimates of Periodograms) is a HUGE field and the standard Fourier analysis is just the start, there many, many other methods that all use the concept of changes/s (=Frequency)

That said, it is of course not the only way; e.g. wavelets are also very popular.
You can of course also look at coherence, auto-correlation etc instead of frequency if you so choose.

Staff Emeritus
2022 Award
I believe you're looking for Time-Frequency Analysis.
To quote the wiki article:
The practical motivation for time–frequency analysis is that classical Fourier analysis assumes that signals are infinite in time or periodic, while many signals in practice are of short duration, and change substantially over their duration. For example, traditional musical instruments do not produce infinite duration sinusoids, but instead begin with an attack, then gradually decay. This is poorly represented by traditional methods, which motivates time–frequency analysis.

berkeman and Pattern-chaser
Pattern-chaser
I believe you're looking for Time-Frequency Analysis.
To quote the wiki article:
The practical motivation for time–frequency analysis is that classical Fourier analysis assumes that signals are infinite in time or periodic, while many signals in practice are of short duration, and change substantially over their duration. For example, traditional musical instruments do not produce infinite duration sinusoids, but instead begin with an attack, then gradually decay. This is poorly represented by traditional methods, which motivates time–frequency analysis.
Oo, yes! That sounds a lot like what I'm looking for. Thanks, I'm off to read the full entry now. 👍

berkeman