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Does anyone know the math for calculating sympathetic resonances? What is it? And is anyone familiar with any instruments where sympathetic resonance is used for quartertones as well as semi and full tones in the tuning?

In case you are interested in a Western instrument with this effected as part of its design: Bazantar played by Inventor (Mark Deutsch).

Well I don't know the answer to you question off the top of my head, but if you're interested in that sort of thing you might check out Fred Carlson's Sympitars if you not aware of them.
Here's a link to his site, there's a video at the bottom with him explaining the instrument.
http://www.beyondthetrees.com/taprootphotos.html

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Jim Watts
http://jameswattsguitars.com

Strings vibrate at their fundamental frequency as well as integer multiples of that (harmonic series). So to get sympathetic vibration, you need another string tuned to one of those frequencies, or with a harmonic matching one of those frequencies.

For example, the B string on a guitar should ring when the low E is plucked, because the third harmonic matches the B's fundamental. Or conversely, plucking the B will ring the low E at its third harmonic (but will not activate its fundamental). And the high E's third harmonic matches the B's fourth harmonic, so plucking one will ring the other.

But if you fret a string, then its whole harmonic series changes, so that changes how it interacts with the others.

Ideally, I think you'd just want one sympathetic string for each of the 12 notes, starting from the lowest frequency of the playable strings. Their 2nd, 4th, 8th, etc. harmonics will cover all octaves, so all fretted notes will be covered too.

If you want less than 12 sympathetic strings, then you could probably get by with 7, and retune them depending on what key you're playing.

But I've never actually built a sympathetic string instrument. Talk to Fred Carlson. He's the master.

Oh and be afraid of the word "overtone". It makes things confusing. Harmonics and overtones refer to the same real thing, but have different numbering systems. First overtone = second harmonic, second overtone = third harmonic, etc. It's all off by one.

So stick with harmonic numbering. First harmonic is the fundamental. Second harmonic is what you hear if you lightly touch the string at the 12th fret, pluck, and remove your finger. Third harmonic is what you hear if you do that at the 7th or 19th fret (which are 1/3 of the way along the string).

Powers of two are the octaves. So 1st harmonic is 1x the fundamental frequency, 2nd harmonic is 2x the fundamental frequency (one octave above), 4th harmonic is 4x the fundamental frequency (two octaves above), 8th harmonic is three octaves above, and so on. Much more logical than the overtone numbering system

And then there are 'partials'. 'Harmonics' are overtones that are exact frequency multiples of the fundamental, 'partials' need not be. Good strings come pretty close to having harmonic partials, but not quite: they tend to go sharp as you go up due to string stiffness. Usually it's not enough to be a problem, but once in a while. Anyway; it's usually better to talk in terms of 'partials' unless you're sure that they're really in harmonic relationships.

I was curious. Under Ottoman musical tradition they use Pythagorean modal calculations for 72 notes per octave set but placing frets only on the 15 places supported by the mode chosen for that song. I know neither the math for their calculations nor the sympathetic ones, but I can guestimate the complexity from your descriptions and what I have read. I suspect that complexity is the reason we do not see sympathetic strings in Central and Western Asia while we do see them in India and a few in Europe.

"I was curious. Under Ottoman musical tradition they use Pythagorean modal calculations for 72 notes per octave set but placing frets only on the 15 places supported by the mode chosen for that song."

To get the fret locations for 12-tone Equal Temperament, you use the 12th root of two, which is about 1.0595. To find the location of the first fret you divide the scale length by the 12th rot of two. That becomes the new length for calculating the location of the next fret, and so on. For 72 fret equal temperament, you'd use the 72nd root of two, which my computer says is 1.00967, and run the calculations the same way. Then you use the fret locations called for in the makam.

For 72 notes per octave, you don't use a computer or a calculator, you use a bottleneck.

One reason that some instruments don't use sympathetic strings is that the sound is not always desired. A piano has more sympathetic strings than any sitar. It also has dampers, and a pedal that controls them, so you can make the sound go away.

Indian music also uses microtones. On a Sitar the microtone are produces by bending the melody string. The sympathetic strings are tuned to the fretted notes. I am not a great sitar player but that is my understanding of things YEMV.

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EddieLee