When playing an artificial harmonic, does it matter where the stopping finger is placed?
Absolutely. Artificial harmonics are natural harmonics that simply use a string that has been shortened by stopping it "artificially," so if you move the stopping finger, you alter the length of the string and the placement of all its harmonic nodes.
- Say that a violin G string has a sounding length of 60mm (measurements in this example are arbitrary, just for the purposes of illustration).
- It will have a natural harmonic at exactly half its string length (30mm), producing a pitch one octave above the open string.
- It will also have harmonic nodes that divide those two halves—one at 15mm and one at 45mm—both producing a pitch of one octave plus a fifth above the open string.
- Now let's say that you stop the string, raising the pitch by a whole step, to A. The new sounding length is, let's say, 50mm. You will no longer find harmonics at the same spots, because the string is a new length. At the site of the original "octave-higher" harmonic, 30mm up the fingerboard (counting from the nut), there is no harmonic any more. Now, the new "half the string length" node is half of 50mm plus the 10mm of stopped string, so can be found 35mm from the nut.*
- Similarly, the "one-quarter-string-length" harmonics are now at
(50 * 0.25) + 10—22.5mm from the nut—and
(50 * 0.75) + 10—47.5mm.
- Shift the stopping finger, even by a millimeter, and all the harmonic nodes re-distance themselves proportionately. If you do not adjust the "touching finger" to match, you will no longer have a harmonic, unless you happen to move the stopping finger to a spot where the touching finger describes a different harmonic node.
So if you're looking for a formula to express the relationship of the touching finger to the stopping finger, it's not complicated: it is the harmonic series. You will always find harmonics in the same places, relative to the stopping finger, that you find natural harmonics on an open string.
There's a fun "special effect," most effective on cello or bass, in which you play an artificial harmonic, high on the fingerboard, then glissando both fingers downward without adjusting their width as one normally would have to do to preserve an interval of a third. As the fingers slide, they mostly produce a "static-y" shriek of descending, fairly diffuse pitch, but along the way they pass through spots in which they describe other harmonic relationships, which ring out, and the resulting effect suggests a seagull's cry.
This effect basically relies on continuously changing "where the stopping finger is placed," and the fact that only a few spots on that continuum produce harmonics.
* This "counting from the nut" business can be confusing and misleading; I just use it to keep the frame of reference constant. A stopped string "doesn't care" how much string there is behind the stopping finger as long as it doesn't vibrate. If you count "from the stopping finger," then you find harmonics not at set distances but at set proportions, ratios of the string length.