Downward Harmonic Series

Edited: February 19, 2024, 6:44 PM · It is not a natural lower harmonic.

We call something a harmonic oscillator, where the force is proportional to the displacement towards the equilibrium. The easiest example is a spring with a weight attached, hanging somewhere.
It has its eigenfrequency, which is the frequency at which it will move freely. This is how mechanical watches work, no matter how much you extend the spring, it will always take the same time to move back to equilibrium and to the opposite side and so on. (Of course there are a few assumptions, mainly that you do not cause any plastic deformation, are within hook range and have no friction etc). That frequency of the motion is the same as the frequency of a note. Luckily, a violin string is almost a harmonic oscillator, it will more or less produce the same frequency no matter how much force you apply. This means, basically, it will keep the pitch no matter how loud or soft you play. As players we do actually know that this is only approximately true, so it is almost linear. Especially the softer strings with less tension can be changed a bit in pitch with the force growing, most easy to create this phenomenon is on the open g string. You can hear the pitch going down as the violin will start to get more silent after heavy bowing.
The frequency on the string does depend on the vibrating length. Basically, the full wave must fit into the string length, so it can produce a standing wave. Imagine a sin function in the strings when you look at the fingerboard from above. The sound will be dependent on the string length and the speed in which the movement can extend through the string, which is similar to the speed of sound inside the string. This is what we basically do when we tune the strings, we adjust that.

Let's go back to the pendulum. Imagine we apply forces periodically now. If you use double the speed of its free movement, you will realize, that the turning in direction of every second of your driving movements will correlate with the turning of the spring in his natural movement. Similar as you do when you use a swing on the playground, this drtives the spring to a good movement and some semi resonancy.
This means basically, you will get a pretty good movement out of the spring.

This is the same thing with harmonics. Anything (mostly) that has an eigenfrequency has other frequencies, that are also easy to drive it with and get a lot of movement. Due to what I described before, there is the tendency to double the frequencies (which is an octave) and some other whole multiples.

Now, in the system you are controlling, there will not be a lower frequency than the lowest there is, which is the base frequency. This cannot be true, there is no "second movement" of the spring that uses double the time, which is what a lower harmonic of first degree would be. This means, in the spring, that the spring will not just move as if I drive it with half the frequency.

The g string will be driven from the g played on d, because it is its natural upper harmonic of 2f. But, it will also vibrate in 2f. So there is no downwards harmonic of your g on the d string, just the upwards harmonic on the g string.
So to make the point stand out more: the g string does vibrate, but not at its "own" note but at the note you are playing.

Another interesting thought of course, as we already talk about it, is, why does it work to vibrate the g on the d string and play open g. The answer is basically the same: your g string has upper harmonics, one of them is 2f, wich is the g on the d string. So if you set this string for this length, it will be driven by the g string and create a sound as well. This string however is inert, so it will keep vibrating if you leave that spot (also if you are very close to the natural frequency it will be resonating even if not hit exactly). This means, you cannot vibrate the lowest g, but the harmonic series.

Our brain knows, that the upper harmonics tell it, which the base frequency is. This was well shown in phones, for example. The phones did not use to go low enough in tone to actually get male voices played correctly. The reduced frequency bandwidth was done to minimize the data bandwidth. So the base frequency transmitted through the phone is basically the same for female and male callers, iIrc the min frequency was in the area of 200 Hz. However, our brain analysis the harmonics and knows, that those belong to a lower base frequency, even if you cannot hear it. So you do know, it's a male that calls. This psychoacoustic effect is why we can hear the g string vibrate when we vibrate the octave, just because we manipulate the harmonic series.

February 19, 2024, 7:12 PM · I think there is an easier way to explain this.

The open G string produces a series of overtones. If its base tone or any of its overtones are "in the air" it will resonate with those. In other wards: Playing the third finger on the D string generates a g which happens to be the first overtone of the open G string. So the G string resonates.

February 19, 2024, 7:18 PM · I'd argue that this is a way to describe it, not explain it :P

Edited: February 19, 2024, 8:04 PM · @Marc

Thankfully I have Google for every technical jargon is see. LOL.

It's mind-blowing to have 2 different A4 on G depending on how loud you play so as to stay in tune. I tried A4 on G and I could bend my pitch 3-4hz on my tuner app above my softest which is so noticeable.

Now I won't see volume the same way as before now that I'm aware how it bends the pitch. As if speed, pressure, and soundpoint are not complex enough. I bow as light as possible when I tune my open strings, I didn't even realize I was taking this sine function into account.

That makes sense, a G on D is 385hz, and since it's the base and the lowest. It cannot make 192.5hz to make the Open G vibrate as 192.5hz and it is vibrating as 385hz despite having an eigenfrequency of 192.5hz. Did I get that right?

I'm just mindblown. I can feel my G on D vibrating while playing open G.

Thanks for this detailed explanation. I learned a lot.

@Albrecht Zumbrunn, yes as the octave of Open G, I really thought before it's making an open G fundamental note.

February 20, 2024, 1:12 AM · While undertones can be explored theoretically and in certain experimental contexts, they are not typically part of traditional musical practice or performance. That's why they are often described as "unusual" or "theoretical."

Edited: February 20, 2024, 2:16 AM · They are very hard to create as well. Basically, what you need to do is, is to put so much energy in the system that it must choose a lower frequency to "store" the amount of energy. Storing because the violin cannot get rid of it fast enough by transmitting it to the surrounding air. However, some instruments also go full berserk instead and just start ringing in every frequency imaginable.
Personally, I never played a piece with undertones, but there is some modern music utilizing it. Very artificial stuff, though, IMHO.
And yes, the pitch changes can be very mind-boggling to handle. This is a reason I don't like starting the Bach double concerto with forte on the open d-string. Also, it is one of the reasons to leave low notes hanging longer in baroque music, so it can stabilize to the right pitch (not the only reason of course, I know the musical theory reasoning as well).
The long open G in Bruch is also challenging for that reason.

February 20, 2024, 4:24 AM · https://youtu.be/oPTt5u681so?si=UNAaow3ncnBp0OJs
To get an idea how it can be used. Sounds like a dying 28k modem to me...

February 20, 2024, 9:48 AM · That is not an undertone. You are confusing two very different phenomena. "Undertones" are produced when you force a string to vibrate at a lower frequency than it wants to, for example with extreme bow pressure. There was a violinist who hit the news years back for refining this technique and using it on stage. "Refine" used loosely, because it still sounds like camel farts.

What you are experiencing is the note on the D string lining up with the first harmonic of the G string, causing it to vibrate sympathetically. Technically every harmonic note will cause this sympathetic resonance, in decreasing amplitude as you go up the series.

February 20, 2024, 10:40 AM · Undertones for me are even worse to play then liston someone else playing them. Everything feels and sounds wrong. I feel like I have to apologize to the violin after it