Tuning and Tension Explained
www.harpsichord.org.uk
Pitch drop after initial excitement is a simple to understand phenomenon: it is caused by strings which are not extended sufficiently. This usually is synonymous with "tension too low", but not always, and increasing tension at the same length (i.e heavier strings) won't help.
The proportion between extension and tension is linear, but the proportion between tension and pitch is logarithmic. This means that the proportion between extension and pitch is also log. Result: the more a material is extended, the more it will have to be further extended to achieve any given pitch rise. Turning it around, the more a material is extended, the less the pitch will rise for any given amount of further extension. This relationship between extension/ pitch is NOT altered by the absolute level of tension, nor by the level of tension relative to the material's breaking load, though since it is hard to separate these factors in the real world, they are often confused. The degree of pitch change per extension increase is simply the related to the amount of ambient extension beyond the zero-load (non extended) length.
What happens is this: while a string is vibrating, it is not straight, but follows a series of arcs (one series for each mode), each arc oscillating back and forth. Naturally, a string is longer if it follows a curve than if it is straight. This means while vibrating, the string has been extended more than when it is not vibrating, meaning also the tension has gone up and the pitch is higher. The larger the vibrations, the greater the extra extension. The pitch is highest just after initial excitement, at the moment when the string has settled into regular vibration at maximum excursion (within a couple milliseconds, usually). As the vibrations die, the string approaches a straight line state, extension goes back down toward ambient length, tension is reduced, and the pitch drops. This occurs each and every time a string is plucked or struck (and bowed, too, as any gut string cello player will tell you).
So why don't we hear it normally? When a string is highly extended, as they usually are, it requires significantly more extension to produce an audible pitch change than for a string which is not extended very much (again, always calculating from the zero-load unextended state). The less the ambient extension, the greater the effect of the temporary extra extension caused by excursion during oscillation, meaning the more noticeable the pitch drop with envelope. Normally, the pitch rise caused by initially high excursions is so small that we don't notice it, though I would imagine many tuners have noticed that the pitch seems to be a tiny bit high during the first fraction of a second of sound. If you are hearing it enough to be bothersome, your strings are too slack - not in terms of tension, but extension.
So how to fix it? You need a string which is more highly extended in the ambient (non-sounding) state. Given a fixed length and material, there is nothing you can do, sorry to say. Thicker strings won't help, nor will thinner, as the tension/ extension the tension part is proportional to diameter and the extension part is material specific. In simple English, it all cancels out. The only way to fix it is either make the scale longer or use a material with a lower Young's Modulus (E). Halve the value of E and you double the amount of extension needed to achieve a specific pitch. This means that the amount of extra linear extension created by any fixed amount of perpendicular displacement is less significant. I suspect your main problem is simply that the scale is too short. Changing to red brass if you are using yellow won't help, the value of E is too similar for both. The only thing that would help is to use a wrapped string, since the extension part of the equation is only applicable to the core wire, which is always highly extended. Actually, that is precisely what wrapped stings do: lower the value of E to something far far lower than is possible with any solid material. Another reason why wrapped strings are used on short scales - it's not just about inharmonicity.
Anyway, the real world consequences of the linear relationship between extension/tension in elastic materials has been totally ignored in organological writings, and it is in actually much more important than either absolute tension or stress levels, in lots of ways. Anyone who wants to get a good understanding of how stretched strings work on musical instruments needs to tackle this issue.
Of course, the big question is, why is the scale too short? Fiddled with, or . . . ? Having just heard Bruce Haynes giving a couple lectures on historical pitch levels last week, me thinks the entire question of scale length/pitch level is in sore need of reexamination. Where are all the northern (presumably iron strung) instruments which must have been made with scales short enough to sound at these higher pitch levels (app. A 465-70)? Perhaps mistakenly identified as brass strung at 415? Just an idea . . .
The instrument under discussion is an Italian. I would also suspect it was designed orignally (albeit copied without an awareness of this) for a much higher pitch, since it was the imported Venetian woodwinds which caused pitch to be so high in northern Europe in the first place.
Extension is quasi-Pfaffensprache for "stretch". If a material has been "extended" to 105%, it means it has been stretched 5% beyond its untensioned length. Getting even more down and dirty, if it is 100 mm long with no tension, it is now 105 mm long. Exactly how much tension is needed to produce this stretch - or turning it around, as in a musical instrument where the wire is stretched between two fixed points, how much stretch is needed to create this amount of tension - depends on the value of E for the wire. The higher the value of E, the more tension per stretch, or the less stretch per tension.
A good way to visualize it is to imagine a wire hung from the ceiling with a hook on the lower end. It is 100 mm long. Suppose you hang 5 kilos on it, and it stretches to 105 mm. If you hang another 5 kilos, it would stretch to 110 mm. Another 5 kilos, and you are at 115 mm. In other words, it would be stretching at a rate of 1 mm for every kilo, or 1 kilo for ever 1%. A wire with a higher value of E might only stretch 1/2 mm for every kilo. Or in other words, you would have to hang twice as many kilos on it to get the same stretch. This is all greatly exaggerated, mind you, just to make things clear. Real wire can only be stretched by very tiny percentages before it breaks, and the forces required are tremendous.
By "ambient extension", I mean the amount of stretch the string is at when it is just sitting there waiting to be plucked. Obviously, it has been extended somewhat, by winding up the tuning pin. Otherwise there wouldn't be any tension on it at all. Plucking it "extends" it even more, beyond ambient extension, first as the quill lifts the string, and secondly as it oscillates along its snaky curvy path.
For those wondering, "E", or Young's Modulus, or the Modulus of Elasticity, is the amount of force required to extend a material to twice its untensioned length. The fact that most materials cannot be extended to twice their length without breaking (rubber bands and bungees being notable exceptions) is of no importance. Since the relationship between extension and force is linear, you can extrapolate from the tiny amount which IS possible. The force must also be taken in proportion to the area of the wire, since thicker wire will require more force to extend it. But the force/area relationship will always have the same proportion to extension. The usual unit of measure is Gigapascals, a unit so far removed from anything in the real world that you can actually lay hands on that I shan't waste your time explaining what the bluhdy 'ell it means.