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You’re right! A tone with such little harmonic content as a sine wave is very rare in the physical world. In fact, harmonics will be created anytime you listen to a sine wave through headphones or speakers because the speakers are also physical objects (just like the Chladni plate demonstration). Great point, Cary. Thanks for watching.

This is a great question, Stuart. I’m not finding any clear explanations online that I can send to you. Although, my theory is that it is similar to room modes or standing waves in that only half wavelength is required.

@@AudioUniversity That suggests it's the sound box doing the pitch adjustment. Makes sense ..... sort of ..... I think.

@@AudioUniversity Boy! Not going to watch over breakfast again* - surefire way to induce a headache before the coffee has a chance! The Wikipedia page on undertones/subharmonics is here: en.wikipedia.org/wiki/Undertone_series, and it's quite interesting. It doesn't mention beat notes, but it does touch on room acoustics (obliquely and in passing), there's also an interesting throwaway comment that says that, in natural musical instruments, _undertones aren't produced at the same time the fundamental is created_ . As long as we're talking about natural instruments, undertones seem to be caused by the acoustic, and/or weirdness in the instrument body, and/or forcing the vibration to be half the frequency of the natural resonance (or whatever value you're aiming for). This make sense if they aren't caused directly by the fundamental, but by a sympathetic resonance, either in the instrument itself or the acoustic. Then there are beat notes to consider, and undertones might be seen as a special case of these. Say you're playing A220, and you add in something else at 330 Hz (a sharp E above middle C) - that gives you the A one octave below (110 Hz) as the beat note (but it will probably sound horrible!). You can try this with a pair of audio oscillators, or simply generate tones in your favourite audio editor... In the 1940s, my late father (who was a designer of synthesizers amongst many other things), nearly got expelled from school with his brother, for sneaking into the school chapel to generate beat notes on the rather fine pipe organ, which had a 16ft rank (of pipes). They found some really low resonances of the building that way, and nearly succeeded in removing the stained glass from the windows! Obviously this is a dumb prank, as you cannot avoid getting caught, but hey. There's another interesting side issue: "wolf notes" in orchestral stringed (bowed) instruments. C.A. Taylor in "The Physics of Musical Sounds" says this is be cause of differences in resonance between string and instrument body. It's interesting because the bowed string is a forced vibration (not like the natural ones of a guitar), so you can force frequencies (notes) with the bow that the instrument wouldn't 'want' to make normally. The implication is that the instrument's body resonance works against the bowed string, rather than with it, leading to a reluctance to make the fundamental, instead settling into another resonance such as 2nd or 3rd harmonic. The 'fix' is usually a damping mass added to the string beyond the bridge (or get a better cello!). I'd suggest that, without some sort of external resonator (e.g. a room acoustic of some sort), you'd probably struggle to get undertones to occur naturally with a guitar (open air/sunny day experiment?), unless you intentionally off-tune a second string. Of course you can digitally process stuff and play all sorts of games with subharmonics, but that's a different matter. *of course I am really!

Thanks, AAAX!

Great point, Jacob. The envelope of the sound is just as important as the harmonic character.

When you create a node in the center, yes. But creating a harmonic at different divisions will create various harmonies.

There are other elements at play, such as the envelope (when the note begins, how long it sustains, and how long it takes to decay). Not only will the instrument have a unique envelope, but I suspect each component frequency has its own envelope as well. Great question though! I've wondered the same thing...

@@AudioUniversity that makes sense, thank you for the reply!

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@@AudioUniversity Yes sir!

I bought it online from a company called Arbor Scientific. www.arborsci.com

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