David, by now you may have recognized me as an interloper among the lute expert, but I'll still give you some numbers from my work with harps. But first we have to make a real point of separation between the lute and other "fretted" instruments (yes, I know the purists fret with gut, and come to think of it sometimes my gut frets at my age).
On the lute and guitar, and other instruments of fixed string lengths that are "stopped" to change notes the problem is a bit different than on the harp, psaltery and other such - where the string is not stopped, each string is its own pitch. But it is not so different as to change much. On the harp the calculation involves the following factors (and I use their terms): Breaking point (tensile strength) Mass (density) Acceleration (a constant that has been variously described to me, it happens to match the 32 feet per second per second of gravity, but I think that is likely a coincidence and the constant has to do with the normal pull by the finger). Percent of tensile strength (a choice, which on the harp has to do with the "pull" and the sound - normally one picks somewhere between 30 and 70 percent depending on the sound one wants). Pitch desired, in cps. As one who has played guitar for well over fifty years I found the formulas, and the empirical comments, to be ridiculous, until I tested them. Whatever the guage of a monofiliment string (the wound strings are different, their mass increases but the tensile strength doesn't - the core guage is the tensile strength) the pitch at which it will break is almost constant, with regard to length. With the lute (theoretically, I must state, as I don't have one) the fixed length means that you have a limited range of base tuning, the open string. And on the lute there are a lot more courses than the guitar, so there is more to worry about. On the harp one shapes the harp for the length of the strings. Now to the string characteristics, as taken from harp strings: Material Breaking strength(psi) Mass(lbs/in3) Nylon 44600 .0383 Gut 52000 .047 Bronze 125000 .320 Steel 325000 .283 Now, there are different nylons, and different guts. Some harps use a "concert gut" which is brighter than the standard (read that to say a higher tensile strength). The same applies to nylon. But an algebraic readjustment of the formula comes to a surprising result. The Nylon, the gut and the steel come to almost the same number when you factor the mass and tensile strength, the brass doesn't. For an example, the psaltery I'm designing would have a C above middle C of approximately eight inches (and unstopped instrument, remember) with nylon, gut or steel. But the same pitch would be four inches with bronze. On my stopped instruments (not a lute yet, but I'll get there) I do use different guages, but that is within a range of the open string tension. The guitar and lute both approximate 30 inches of vibrating length (a bit more, a bit less depending on the instrument) At that length they can only go "so high", or so low on monofiliment But guitars use wound strings for the bass, and I assume lutes do. C below middle C on a harp would require a five foot long string if one didn't put a wound string on it. But remember, the gauge quoted on the label is that of the wound string, not of the underlying core which defines the tensile strength, but not the mass. Hell, got to get up in the morning, long drive to my annual TigerTones reunion, let me know if you lutenists want to hear more about strings in their natural state (unstopped). Best, Jon
