Thank you @nate for these clarifications. I observed the same phenomenon on my spectrograms, indeed. Let's not forget that if we make a string vibrate, the sounds perceived are those produced by the entire instrument, that is to say, air and wood. By comparing different dulcimers built in different woods, I noted a whole variety of responses, for the same frequency, depending on the wood used. These variations affected both the number of overtones perceived, their amplitudes at attack and decay, and their sustain.

Thank you for taking the time to articulate your results dulcidom! I have been playing around with these ideas quite a bit. I have a one string dulcimer that i have tuned to exactly 100 Hz and a weighted pendulum that i have been moving down the length of the string and then observing the results in a spectrum analysis software. One observation is that on an instrument like a dulcimer, the overtones that sustain are not always the overtones with the sharpest attack. If one were to look at peak amplitude of frequencies immediately after the "attack" i have found them to be different from the ones that sustain the longest. 

To put it simply and summarize my previous post, it seems that the sound in the middle of the string is "warmer" and "mellow" because it is energetically dominated by low odd overtones that mimic the perfect major chord (#1 = root, #3 = fifth to the octave and #5 = major third to the superoctave). This perfection is gradually broken going towards the bridge, activating less consonant overtones of higher frequencies....

Thank you for this very interesting discussion.

Excited by the question,I took some measurements on the melody string of one of my dulcimers with an acoustic spectrometer. I successively plucked the string at different points along the VSL, starting in the middle (fret 7) and gradually moving toward the bridge.

The measurements showed that all the notes emitted had exactly the same scale of overtones (I measured up to the 27th...).Qualitatively speaking, none of the resulting sounds was purer than any other, wherever the string was plucked.

The only visible difference between all these sounds was the energy associated with each individual wave. I read somewhere in this thread that the octave (second overtone) might be responsible for the mellowness of the note obtained in the middle of the string, but this overtone was actually almost nonexistent at this plucking point, where odd overtones #3, #5, and #7 dominated the vibrational energy. The octave only appeared when moving toward the bridge, around frets 11-12, and disappeared again toward the bridge.

Near the bridge, too, the vibrational energy was distributed between overtones #3 to #8 with higher frequencies, in keeping with the brightness of the resulting sound, and its slightly acidic character, given that the higher overtones become a bit dissonant (this is quite noticeable for #7).

If that helps...

Really well said John.  
I commend and thank you for contributing to the scientific understanding of string physics. The words may be subjective, but this thread has compelled me to try to understand the "objective" side of tone a little more.

Since we seem to agree where a string is plucked does make a difference, let’s not forget that the main thing is to use the varying sounds to make better music. 

Adjectives for sounds such as “warm”, “bright”, and so on are subjective.  How we perceive and describe things varies.  Discrepancies among us should be met with tolerance. 

There may be a reason why plucking somewhat away from the center might be preferable to plucking exactly in the center.  The mathematical analysis predicts that as the pluck point is moved away from the center, the overtones get stronger, but not uniformly.  The overtone that comes in strongest first is the octave overtone.  To my ear, octaves reinforce (for lack of a better word) a pitch.  Try playing D (open bass string) and then Ddd (open bass string, 3rd fret middle string, open melody string) to get a sense of this.

Thanks for the discussion.

This topic has got me doing a lot of research and really appreciating how far beyond me most of this stuff is to try to wrap my head around. After a few hours playing around in an audio spectrum visualizer, Im really starting to appreciate just how unreliable my ear is.

Such fascinating stuff 

I think overtones are associated with our perception of depth. An example of an instrument with almost no overtones is a tuning fork, and an example of an instrument with many pronounced overtones is a cello. 


I also think what Dusty mentioned about the string being floppier near the middle may be the main reason that the tone is so different near the middle of the string. I have noticed that the actual pitch of the string wobbles a lot more when I pluck near the middle. Though very unscientific, I just plucked my string as hard as I could right at the bridge and noted that the frequency wobbled in a range of 1 cent. I plucked as hard as I could at the middle of the string and found that it wobbled in a range of about 4 cents. Since the actual string is producing a larger range of frequencies when plucked at the middle, this might be another factor for why the tone is "warmer" near the midpoint of the string.

I've been wondering a lot about this stuff for a while. I think today is the day I make string striking pendulum and start experimenting!

To my ears, a note does sound more 'clear' when plucked closer to a bridge, but it also sounds more hard and 'tinny' (to me), while a note plucked closer to middle of string sounds rounder and warmer, but I'm not necessarily hearing 'overtones' in it, just a little more depth. The 'tinny' quality strikes me as more of an overtone, if anything. With any description within the human senses, I think there is a certain amount of subjectivity based on personal perception of sound, taste, touch, etc.

Thanks so much, John for that explanation.  Basically, you are defining purity of tone as a minimum of overtones.  That seems logical.

But it also raises a question which I asked earlier, which is how the mathematical definition of purity of tone corresponds to what we subjectively hear.  It certainly seems to our ears that the brighter sound of notes plucked closer to the bridge is more accurate or pure than the warmer sound of notes plucked closer to the center.

My question about fingers and pick was, well, kind of stupid honestly, since clearly those are variables you would want to hold constant if you did a physical experiment and something you would not choose to address in a theoretical simulation.

But there are real-world issues that muddy the cleanliness of your mathematical model.  In your first diagram of the inverted V, h represents the extent of the pluck, or how much the string is pulled before being plucked.  But there is a lot less tension on the string in the middle than there is on the ends.  To pull that string to the same extent by the bridge would result in a much more violent release, increasing the volume, certainly, but also, I assume, changing the nature of the string's vibrations.  I think you might need to add another factor--string tension--to this study and adjust h in that diagram to keep tension constant. Does that make sense?

Thank you for taking the time to explain John. I have to admit, I find the results counterintuitive. If a "purer" tone means less overtones, and an instrument like a flute is an example of a very pure tone with few overtones,  it seems to me that I get a "purer" tone while plucking near the end of the string than I do close to the middle.

I've attached a response for Dusty and Nate.  Others are welcome to dig in.  It's a little lengthy, so I had to put it in a zip file which you should find attached.

VibratingStringExplanation.zip  •  67KB

I've heard it said that the closer to the midpoint, the more freely the string can excite overtone frequencies. The way it was explained to me is that when the string is divided into two lengths on either side of your plectrum, whichever length is shorter will limit the overtones of the longer one, and therefore the greater the difference between the two lengths, the more the fundamental is emphasized, whereas the closer the two lengths are to equal, the more overtones are emphasized.

If I understand you, @johnr, by "middle" you mean at half the string's length, or the octave (7th fret on a dulcimer, 12th fret on all those chromatic instruments).  We would normally say you get a "warmer" or "softer" tone there than you do towards the bridge, where the tone is usually described as "brighter."  So the warmer tone corresponds to what you describe as "purer" based on the sound waves.  By "purer" do you mean that there are fewer waves that bunch together to make a note?

How did you pluck or strum the string?  Whether you use a plectrum or bare fingers also affects the tone in ways that I assume you could measure.

In response to Dusty's "How Do You Measure Tone Mathematically" - I would call it more of a description.  A very simple, perfectly pure tone for a specific note would be a wave.   Real tones are actually built from a bunch of waves which are related to a basic wave which corresponds to the note.  The relative strengths of those tones are the basically the heights of the waves.   Mathematically these satisfy a partial differential equation which is known as (TA DA) the vibrating string equation (or wave equation).   The remaining part is how these waves get going.  That's what happens when you pluck a string.  What I was able to show, mathematically, is that where (near the end or near the middle) matters.  Closer to the middle gives a purer tone.  I think that's what dulcimer players intuited a long time ago.

@johnr , that is 'too cool for school'!  Something very much to be proud about.

And thank you for supporting FOTMD as well... so kind of you!  

Congratulations, that's awesome. Since I likely wouldn't be able to interpret the paper even if I could access it,(math and I are like nodes and anti nodes ) are there any insights you could share about the ideal place to strum, based on your findings?

Congrats, John.  I was able to locate a summary and your bio, but Taylor & Francis won't let me see the whole text.  My library has a four-year delay for full-text articles of that particular journal. That's OK. I learned a little about you and can see from the summary that I really wouldn't understand the text anyway.  I may request it through Interlibrary Loan just to add to the dulcimer library.

How do you measure tone mathematically?  It seems like such a subjective quality.

Very cool, @johnr!

Everyone - this is a thank you, a brag, and announcement of a mountain dulcimer picture in a place you wouldn't expect - Mathematics Magazine.  After reading and watching many of you on this site, I was inspired to investigate an aspect of the vibrating string partial differential equation.  Thanks!  This resulted in a paper "What I Heard from the P.D.E." which has just been published - Mathematics Magazine, Vol. 98, No. 1, February 2025.  In the introduction, there's a picture of my dulcimers.  Most college libraries have subscriptions to Mathematics Magazine.