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!  

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Site Owner

Those irritated by grain of sand best avoid beach.
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I've always been fascinated by that question. As players we intuitively know that plucking at different distances from the bridge produces different tones. It makes sense to me that this could be described mathematically...just maybe not by you or I...

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.

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Dusty T., Northern California
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As a musician, you have to keep one foot back in the past and one foot forward into the future.
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updated by @dusty: 03/29/25 04:15:13AM

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.