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.
I play semi-regularly with a small/informal (work) jam band. Not only is my dulcimer impossible to hear over amplified instruments, but I feel their music resonating in my dulcimer more than my own strings.
I've seen those stick-on piezo pickups, but my question is would those even function in an environment where I'm feeling other instrument's in my own soundbox over my own strumming? And if not, what should I look for/ask a luthier to do for me?
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!
Thanks so much for your input.
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.
Someone's idea of an upgraded McSpadden kit.
It resmbles a McSpadden teardrop dulcimer, but the use of 5 pieces on the back sets it apart. Also the red color is different, and the type of tuners. It may be a homemade copy of a McSpadden.
Funny...there were 2 McSpaddens in the same lot. This one does not have the stamp though.
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.
Looks like a McSpadden. They often had their name stamped below the bridge area. Most likely a kit.
I was wondering if any of my friends recognize this builder. There is no tag inside the dulcimer so I'm thinking it is a homemade. That scroll head is pretty distinctive, though. Thank you for your help!
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.
FOTMD is where IADD was conceived. 
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.
@robin-thompson Yup. I'd say you folks got the ball rolling, and now it's gathering some momentum. I tell everybody that the dulcimer is a more logical first stringed instrument than the guitar, so it makes good sense.
@ken-longfield Thanks ... and you're very welcome!
Sam, thanks for the information about Brandi's song and for the link to Tim Hanseroth-- I'd watched that some time ago yet hadn't looked at it lately.
Participation in IADD came from different countries-- Ireland, England, Australia, US, and, perhaps, other countries of which I'm unaware. So glad to see many places represented!
Sam, that was a delightful video segment. You and the reporter did a fine job. Thanks for your help in promoting IADD.
Ken
"The dulcimer sings a sweet song."
Hi, friends - Thanks for bringing up my TV appearance Friday! I'll admit that I'm normally not a big fan of "Hallmark holidays," but dulcimers are a cause that deserves the publicity. I didn't realize its origin was so closely tied to this thread and these people!
@robin-thompson Brandi and the twins are listed as the three writers of the song. It came out of the pandemic and is about Brandi's relationship with her wife and daughters. (I also see the song as her version of Big Yellow Taxi, except that the relationship is a strong, positive one.) They decided not to use the dulcimer on the record, feeling it would be "too Joni, so she plays the riff on guitar, but I had to see video to convince myself it wasn't really a dulcimer.
Here's an IG video of Tim playing a bit.https://www.instagram.com/timhanseroth/reel/CkM6mWlsjMj/
@dusty Thanks - I got a kick out of their "Dulcimer expert" label, too!
By the way, the reporter in the news segment was very good. Before we went on, she asked me the pronunciations of Appalachian, dulcimer, and my name.
I think Ken is correct that the fretboard overlay appears to be walnut. I believe the body and sides are ambrosia maple with a light (probably oak) stain. You can see the small pinholes and streaks typical of ambrosia maple under the stain. The top appears to be curly maple.
Ron
Thanks so much for this insight! I got some strings locally and strung it up yesterday and it sounds amazing! :)
In the scientific discipline of organology, the study of musical instruements, the dulcimer is by definition a "plucked box zither" and therefore, by definition, if the fretboard extends beyond the body of the instrument, it can no longer be considered a dulcimer and would therefore be a lute. That being said, in layman's parlance, many instruments which do not fit the organological definition are considered by many to still be dulcimers. Historically some dulcimers were played with a bow rather than plucked, although Ken Bloom's "bowed dulcimers" resemble a gamba more than a traditional dulcimer. Builders like Gary Gallier extend the fret board past the body which gives the instrument a more guitar like sound, but it plays exactly like a traditional dulcimer. Chromatic fretboards are way older than some folks seem to think, and do not affect whether the instrument is considered a dulcimer from an organological standpoint. Personally I think a picking stick goes too far and really can't be considered a dulcimer at all. The "dulcijo" or "banjamer" is a hybrid and also really must be considered a separate instrument since the resonating surface is no longer dependent on being a "box", but is now a membrane instrument like a banjo, rummelpot or a drum.
I think Ken is correct that the fretboard overlay appears to be walnut. I believe the body and sides are ambrosia maple with a light (probably oak) stain. You can see the small pinholes and streaks typical of ambrosia maple under the stain. The top appears to be curly maple.
Ron
Happy day after IADD, y'all!
@dusty Thanks for the kind words! We made our usual put-someone-to-sleep type of tune only recorded with a camera. (We hadn't made a video in a year or two because our uploads took many hours. Our cable internet is better now.) I love your Freddy Fender cover! In my mind's eye, I see him with his thick dark hair and wearing a purple suit. So cool you covered one of his songs!
@sam-edelston Such a great spot, Sam, and I'm glad you got to promote your album (it's a very cool album)! I'm guessing one of the Hanseroth twins was the composer of the Brandi Carlile song. It was a nice one to end the spot.
@steve-c Many thanks for sharing the link for Sam's interview!
@sam-edelston is a rock star, for sure!
I like the title they give him on TV: "musician and dulcimer expert." I want that on my business card.
Sam Edelston on tv for International Appalachian Dulcimer Day! : https://www.wtnh.com/video/celebrating-international-appalachian-dulcimer-day/10580088/
That's beautiful playing, Robin. Your sensitive approach to the noter is second to none. Peace and goodwill to you, too. 
Well I was able to make a video using Clipchamp for the first time, but YouTube only let me upload it as a short. If it ain't one thing . . .
https://youtube.com/shorts/rKRztEXOx8E?feature=share
Many thanks, @katiewaller! I hope you've had a chance to join in on the celebration of IADD!
@dusty That's just cool you were the mystery guest for Send in the Music! I'm sure your play was wonderful and enjoyed by all your audience!
Happy IADD, everyone. I tried to record a video last night using OBS studio, but the audio came out distorted. I'll try again using different software a bit later.
However, I was the mystery guest at Pat Clark's Send in the Music today. She will post the video soon, if it's not already up. Today is also the 5th anniversary of Send in the Music, so lots to celebrate. I was kind of nervous and my playing was pretty spotty, but here was my set: Reel de l'Oiseau Bleu, Cheticamp, Linstead Market, All in a Garden Green, Lady Gethin, Fishin' Blues, and an original, Safe Haven.
The lower instrument in my photo on the left is one I made a few years ago. The VSL is about 15 inches and it has a standard string set. It is tuned to DAA, but an octave above normal - A = 440.
It has a nice sound, but I have hardly played it. I had in mind that t might be used in an ensemble for adding accent on a second verse, or whatever. I haven't had an opportunity to try that with our group.
Apologies. My wife and I are not tech savvy and can't seem to attach photos. But, your comments have been very helpful. Thanks again for your kind assistance.
@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! 
This sounds more like an epinette or maybe even some other instrument than a mountain dulcimer (a ukelin, a psaltery?). The length of the whole instrument is 17"... that is reeeally short. And we don't even know the scale length. Is there any possibility you can attach a photo if it here?
VibratingStringExplanation.zip - 67KB
