Mountain Dulcimer Picture in Mathematics Magazine
General mountain dulcimer or music discussions
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?