Comparing intervals to harmonics
You may already know about the harmonic series. When you play a pure sine wave on a synth and look at it on a frequency analyser, you should see a single spike at that sine’s frequency (assuming your synth isn’t replicating an analogue sine tone, which is often the case). Next, change the waveform to a saw wave, and you’ll see many more spikes appear above that spike.
These harmonics are governed by natural science. The first harmonic, as seen in the sine wave, is your ‘fundamental frequency’ – 440Hz for note A3. The second harmonic appears at twice that frequency (880Hz), the third at three times that (1320Hz), and so on. Each harmonic is separated by the frequency value of the fundamental. So the first few harmonics of a wave with a fundamental of 48Hz are 48, 96, 144, 192, 240, etc. Now compare these frequencies to the actual notes we play, and you might notice there are some discrepancies. Let’s take A3 (440Hz) as an example. Its second harmonic is A4 (880Hz) and its third harmonic is 1320Hz. That’s close to the frequency of the note E5 (1318.5Hz), but not exact.
The next harmonic is A5 (1760Hz), but then things really start to go out of whack with the next harmonic, which is 2200Hz – the closest note to this
C# is 6 (2217.5Hz), which is quite a discrepancy.
What’s the problem? Well, if you’re playing an A3 note along with an E4 note, you’ll get a 2217.5Hz from the A note and a 2200Hz harmonic from the E4 note, and these will clash slightly with each other. There’s a system of note tuning called ‘just intonation’ that aims to eliminate these clashes, but it comes with its own set of problems.
For now, it’s enough just to know the roots of where notes and scales come from, but if you want to learn more on that subject, there’s plenty of useful info online.