What is the relationship between where and how a vibrating string is activated?
My understanding is that each string on a musical instrument/chordophone has a "sweet spot", such that when a string is activated at that point, the "best" sound is produced.
Further, a string can be activated by a hammer (piano), bow (violin), tangent (clavichord), or plectrum (harpsichord, guitar).
For a given (possibly "ideal") string, is the "sweet spot" fixed, or is its position affected by the activation method?
What is the basis for that relationship (or lack thereof)?
The motivation for the question comes from imagining a keyboard instrument in which the action can be manipulated so that notes can be played with a hammer, plectrum, or tangent. Would each mechanism need to strike at the same point, or could the strike-point vary?
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My understanding is that each string on a musical instrument/chordophone has a "sweet spot", such that when a string is activated at that point, the "best" sound is produced.
As far as the violin goes your understanding is wrong. Putting to one side what is meant by "best", the three variables for violin playing are:
Bow speed
Contact point
Bow weight or pressure
Juggling these three elements on any particular string produces different effects.
Generally speaking, close to the fingerboard you need light pressure and a fast bow. While close to the bridge you need a slow bow and more bow pressure. This is complicated by the length of the vibrating string. The more you shorten the string by placing a finger on it the closer to the bridge you have to play AND the less pressure you should apply.
Then there is the question of the way you (or the composer) wants the note to sound. The same note in different circumstances can call for a light, fast bow close to the fingerboard for the "best" sound in one part, a slow, firm bow close to the bridge in another and a moderate bow halfway between in yet another.
It is complicated, far more complicated than you would think.
According to this source there is a physical reason for where a piano wire is struck and the best location for plucking would be different:
After many tests throughout the history of the piano, it was determined that
the best strike point is between 1/7 and 1/8 the length of the string. In general, it can be verified that if the point where a string is plucked (not struck) coincides with a node of any one of the vibration modes, that mode will not be excited. The most intuitive case is the fundamental of a string fixed at both ends. Its nodes are at the ends, which means that the greatest excitation of the fundamental will happen if the string is plucked exactly in its center, that is, at the antinode of the first mode. As the string is plucked further from the center, it vibrates less, and it is impossible to excite it by plucking right at an end. Likewise, if a string is plucked at 1/7 the length, the 7th mode is not excited, along with its multiple integers: 14th, 21st,.... In music, this phenomenon can be an advantage, since the 7th harmonic is dissonant to the tempered minor 7th. This fact has been utilized as a justification for the choice of strike point in the piano, and it is still
affirmed in some relatively recent articles and books.
The report cites the following books for the paragraph I quoted above:
H.F. Olson. Music, Physics and Engineering. Dover, 1967.
C.A. Culver. Musical acoustics. McGraw-Hill, 1956.
E. Good. Giraffes, black dragons and other pianos. Stanford University
Press, 1982.
This is an interesting question. I'd refer to texts by Fletcher and Rossing on the subject as they are world leading experts in the physics of musical instruments. I can only tell you what I've learned from guitar playing, specifically classical guitar.
You should know that the brightness of the note produced by the string depends on where you pluck it (I am referring only to guitar and not hammered or bowed strings). So it is hard to say that any spot is "sweet" since that is somewhat subjective. If you wanted warm tones, al la Wes Montgomery, you would be wise to pluck at the mid point as that will accentuate the fundamental and kill a lot of the harmonics. If you wanted a very twangy tone you would want to pluck near the bridge as that will excite a lot of high frequency overtones and with considerable amplitude. So it would seem like you do NOT want to stay in one spot if you want variety and control of your tone. But who's to say which of those tones is "sweet"?
There are other factors to consider. By experimenting with the attack placement one may sacrifice other qualities like sustain and volume. So there is a trade off and I am not sure anyone has really studied this is detail but if anyone has it would be in Fletcher and Rossing's texts (if not originated by them). Various schools of thought are split on this in the classical guitar world. I recall Pepe Romero's book teaching that the strings should always be plucked at the same spot, just behind the sound hole in the direction of the bridge (I am pretty sure it was Romero, but could be Parkening) and that any other variations on tone could be achieved by varying the attack angle of the nails, pressure applied to the string, etc. In other words his method supports the idea that one gets overall superior sound in one place and that other variety can be achieved by controlling the attack parameters. However, not everyone agrees with this. I have seem video of Julian Bream where he will sometimes pluck over the finger board. It is also quite customary for Flamenco guitarists to play fast runs by plucking closer to the bridge but this is not for tone as much as the string feels stiffer there making it easier to bounce the fingers off the string, in contrast to feeling like a loose rope or wet spaghetti when plucked further in. Based on this you can see that there are other factors than sound that contribute to these decisions.
I cannot say for sure that Romero is correct (or the comment I have attributed to Romero) but in my experience it sure seems to be. When the hand drapes over the strings the fingers will lie across them diagonally, each finger touching the string at a slightly different place. This very fact makes it impossible to achieve consistency in placement, especially when playing chords. Perhaps they are in a small enough neighborhood about the sweet spot that it doesn't matter. My opinion is that this is true and it is a standard part of classical guitar training. But I have not seen objective data on this and would deffer to that.
The above does not likely hold as much for the electric guitar. While the basic string physics is all the same the amplification, sustain, and to some degree the tone, can all be controlled by electronics. With just a plain amp you can still hear the variety of tones generated by pick placement but the other issues that exist in generating a reliable note on the classical just do not exist for the electric. At the extreme I would guess that a large enough effect rack could make attack precision an unnecessary skill.
Hrmmm.
As a guitarist, where I pick gives me control over expression. To some extent, pickup placement replaces it for electric instruments, but even then, picking close to the bridge has an effect. I'd say that over the soundhole is the "sweet spot", but seeing that it's fretted, that moves, but even then, the unsweet spots can still be pretty sweet.
As a person who has progressed past the screech and is now a very bad fiddle player, is very much toward the bridge, but the need to have a waist so that the bow can angle confounds this. I'm sure there's other issues that I'm not experienced enough to know. Similarly, I'd expect that hammer placement on pianos, also pick placement on harpsichords, is more about where they can be mechanically placed than where they're musically optimal.
So I'd guess there's enough wiggle room for a few methods, depending on how high a note you're hitting. I wonder about the actuation, but that's not the question.
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