AN ENLIGHTENING EXPERIMENT
In my posts above you will find some “strange” data from Lutz:
.026" 80 PSI
.024" 80 PSI
.022" 80 PSI
.020" 110 PSI
.018" 110 PSI
.016" 120 PSI
.013" 120 PSI
This sequence shows two mathematical oddities:
It is unlikely that a breaking point is rounded to the next 10 PSI: this shows that the measure is coarse.
It is unlikely that the breaking point changes abruptly between .,022" and .020" and then from .018" to .016". This also shows that these values are coarse.
The independence of breaking point from string gauge (with the exceptions I have already noted) is easy to demonstrate from first principles, but it is also very illustrative to prove with a very interesting experiment I have performed quite a few times. You need:
a sonometre, in this case a very simple one: just a plank of solid wood or similar material long about 110cm, with a harpsichord-type hitchpin on one end and a harpsichord wrestpin on the other end, carefully placed exactly 1 metre apart (a tolerance of 1mm will suffice to guarantee accuracy to 1/1000, obviously much more accurate than the errors in values supplied by some makers.)
an electronic tuner showing exactly the pitch in Hz a note has, and able to respond quite quickly.
a formula for PSI, another for kg/mm2 .
For a practical harpsichord test:
Take a string coil and make an eye on one end, the way you usually do for a harpsichord.
Install the string in the sonometre as you would in a harpsichord.
Calculate approximately the breaking point pitch from the string specs you have.
With the tuning fork tune the string up until you are about a fourth below breaking point.
Now carefully (and looking all the time to the electronic tuner) tune the string up very slowly until it breaks. Since a semitone is equivalent to 5.9% of pitch, to have an error of measurement smaller than 1/100 you have to ensure that you take more than one second (the time for the electronic tuner to react) for every 1/6 semitone, that is, take at least 6 seconds to raise the string by a semitone. This will not be very accurate, and we expect errors of about 2% in the final result.
Once you know the pitch at which the string broke, a formula (have to find it somewhere )will let you know the breaking point:
To get really accurate results, repeat the experiment at least 3 times. Of course, this is a bit expensive.
The above will give you the breaking point of a string as recently installed on the harpsichord. In the long term, string material undergoes fatigue at the eye (obviously) and at the wrestpin, where the string is turned from straight to curved, and therefore in practice the breaking point is lower, which is why they recommend the well known tolerance of 10-20% according to the alloy.
For a scientific measurement of the breaking point, you can perform the experiment with the following modifications:
a) Use a piece of string with no eye on it.
b) In the sonometre instead (or at a side) of the hitchpin, have another wrestpin.
c) Best use modern zither pins (larger diameter than traditional ones) or even a modern piano pin, even wider in diameter.
This is what makers (or resellers) should always do, and would produce a value that can be expressed with confidence in integer PSI, not in tenths.
Now repeat the experiment with any alloy in any three successive gauges (not the thinnest ones), and you will get exactly the same result!
Edit: the formulae required are obviously just a reversal (secondary-school algebra) of Mersenne’s one, with parameters to cater for different measurement units.
Interesting comments about string guages, and the (approximate) independence of breaking point and string diameter.
When restringing or building replicas of historical instruments, of course, the main issue is to figure out what the original builder was using, and indeed, one generally finds that the stringing schedule roughly tracks the breaking point, but with a “margin of safety”. It is true, one doesn’t want a harpsichord that is always in danger of breaking strings because they are tuned too close to the breaking point; but there is another important consideration: tone quality. This concern might apply more to those of us who experiment with new instrument designs (a tiny minority, it seems), but it seems worthwhile even when simply reproducing what some ancient builder has done, to understand all the factors that went into the design they created.
It’s not just about preventing string breakage, and generally speaking, strings don’t necessarily sound “best” when tuned as close as possible to breaking. A corollary of this would be that the aim should not always be to simply minimize inharmonicity.
In my own very limited experience, I designed an instrument with range and stringing similar to a harpsichord (not plucked, however, but that’s a different story). I did substantial theoretical work before building anything, and I had what I thought would be a good stringing schedule figured out, purely from “the numbers”, and from following this (mistaken) notion that generally I should try to stay very close to the breaking point. However, I had to do a significant redesign before building commenced. I finally got the real strings in-house so that I could begin acoustic experiments, and I put a length of string onto a “sonometre” as described by Claudio (or I called it a monochord). I found that there was a tremendous variation in tone quality, as I tuned down away from the breaking point; and some of the lower-stress tones sounded better to me, than the maximally-stressed tone. Of course, there are tradeoffs and there is a limit before other disadvantages of a too-loose string start to come into play; and also, I was simply pursuing what I felt was a pleasing tone from a new instrument design, I was not trying to recreate any particular existing tonality, so in that sense I had complete freedom. I found that to my taste, at least for this type of instrument, I preferred the sound of strings tuned “two whole tones below the highest viable pitch” (i.e., a little further below the breaking pitch itself).
I had planned to build an instrument with four octaves starting at E, but I ended up with four octaves starting at C.
I would recommend to anyone interested in the topic, to make a simple monochord and experiment with string tuning, paying attention not only to behaviour near the breaking point, but also explore the “tonal sweet spots” well below the breaking point.
BREAKING RISK RANGE. Common wisdom from ancient times is that instruments sound better when strings are—within safety limits—near to breaking point. In our spreadsheet we have computed breaking risks for Taskin’s historical stringings and observed that, in all the [Taskin] instruments included above:
Iron strings have mostly risks between 65% and 80%, confirming “common wisdom”.
Yellow brass strings behave differently, because they have very frequent changes of wire size: as we proceed from bass to treble, risk increases from about 50% to about 80%. This still follows “common wisdom”.
Red brass has risks between 33% and 55%. 18th century makers could easily have used red brass in the 8’ up to A with no significant risk, yet they stopped at D or Eb at most, surely because they found that yellow brass sounded better from Eb upwards.
Back to your comments, Benjamin, the above risk limit 80% is in tension, therefore proportional to the square of the frequency. You wrote “two whole tones”, and for the sake of this discussion it is a good approximation to use ET. Two whole tones are four semitones, thus the interval ratio in spreadsheet formula is approx. 1.059^4. More accurately, this being an ET major third, the formula is 2^(1/3) = 1.259921.For the tension distance from breaking point we have to take the inverse squared, therefore 1/2^(2/3) = 63%.
For iron this appears to be a low tension indeed, but is good value for brass. Note however that my average tension curves for Taskin’s stringings are NOT constant in “semitones” (thus in breaking distance in percentage), but vary across the instrument range, as shown in my webpage. Regardless, and especially in a special kind of instrument, the best tension is to be found for each particular instrument.
We need more musicians like you, Benjamin, doing real-life experiments and drawing good conclusions.
Forgive my delay in replying. I will see what I can do about updating the link. My webmaster art conservator daughter had recently had to leave me to handle my website alone and that is on today’s worklist!
)will let you know the breaking point: