| I am sure you have all heard that a sharpening lever is really nothing more than a fret
that stops off a string at a given point to achieve a given intonation. I wonder if you have ever considered why the
harp is the ONLY fretted instrument that needs the frets to be tuned. But is this true, or is the need for tuning
inherent in the design of the levers themselves?
For any levered string, there are 4 points that must be considered when considering the intonation of the lever.
First is the bridge pin, then the mounting for the lever, then the lever stop off point and finally the point where the
string enters the outer string rib. Of these, the bridge pin and the mounting for the lever are fixed and immutable. The
only two points, which can possibly change, are the lever stop off point, and because of string tension on the
soundboard, the overall vibrating length of the string. We are usually told that it is this last which causes dis-intonation
of the lever. BUT IS IT?
Certainly when the harp is first strung and brought to concert pitch there is significant bowing out, and the overall
vibrating length of the string is reduced. These initial adjustment changes are only transitory and within two or three
weeks of being kept in tune, the harp comes to a state of equilibrium where the Vibrating lengths of the strings remain
very constant.
At this point, I would hope that you would not take my word or the word of any harpmaker or technician when each of
you has available what must always be considered the final authority on harps. That is your harp. Actually I implore
each of you
to develop the habit, where ever possible, of asking your own harp. Here, you might pick at random one string from
each octave, write down the number of the string, then measure it to within 1/16 inch from the bridge pin to the first
point of contact with the outer string rib. Write down these string numbers and their lengths, put this information in a
dated envelope then place this in the harp. It only takes about 10 minutes to do this. Several months down the line,
when you have a few minutes, re-measure those strings and see if there is indeed any change in the vibrating length.
To get an idea of how much movement must occur in a soundboard to generate a given change in intonation, let me give
an example of a given string going out of tune by a given interval. The interval I’m going to use is the so-called
syntonic comma. For a strict definition of this, you may look it up in any major Dictionary of Music, but suffice it to
say that it is an audible, but barely so interval. About what you might expect in the pitch change of a moderate vibrato
of a violin. Or mathematically, about 1 / 9.3 of a whole tone, or 4.65 commas for each even tempered half tone
***Now, consider a string 31.784 inches long that has a lever stop off point 1.784 inches below the bridge pin. (The
correct 12th root of two distance for even temperament) leaving a sharpened vibrating length of 30.0 inches.
To raise this by one comma, the vibrating length would have to be diminished by about .370 inch. That is nearly 3/8 of
an inch. And that’s not all, since the string enters the soundboard at an angle of somewhere near 35* the soundboard
would have to be pulled out by secant 35* X 3/8 inch = nearly ½ inch. Much more than a comma out of tune, and you’d
be picking up pieces of soundboard off the floor.
Of course, for the lever to go flat by one comma would require the vibrating length of the string to increase by
slightly more. Here, the increase would have to be .375 inch But it is really hard to visualize how a soundboard under
hundreds of pounds of upward pressure could possibly move down to increase the vibrating length of the string. I truly
doubt that levers going out of tune can be attributed to harp bodies that exhibit symptoms of either anorexia or
pseudocyesis.
What then is it that makes the lever go out of tune? To answer this, it must be understood that when
initially placing the lever, the stop off point must divide the string into a vibrating length and a non-vibrating
length of a very precise ratio. This ratio is far more affected by any slight movement in the position of the stop off
point than by the diminishing of the vibrating portion of the string |
.
To calculate the change in the stop off point for making the intonation of the lever one comma sharp, one need only
apply the same ratio as found when computing the same interval (one half step plus onene comma) as a result of the
bowing of the soundboard.
This change in the stop off point turns out to be only .021 inch.
At this point, we might well want to ask: "What is it that can change the adjustment that the lever has been set
at?" One word will suffice, Vibration. Consider the sharpening lever on the Middle A. Consider it in the
"ON" position. As such it is being lightly tapped 466 times per second by the moving mass of the string whose
movement it is reflecting. Since there is no positive lock on its adjusted position, is it any wonder that it can be
moved a few thousandths of an inch over period of time and dependent on how often it is in the On position? Of course,
as the frequency increases on the higher notes, so too does the vibrational pressure on the tuning mechanism of the
lever, and Newton’s second Law will have its way.
If, as it seems from the above analysis, that it is the very fact that the levers are tuning adjustable that they go
out of tune, why do
harpmakers and lever manufacturers have the adjustable feature on them?
The answer to this is threefold. First, it would be necessary to fully season the new harp strung and kept at concert
pitch for 2 to 3 weeks before even considering mounting levers to the harp.
Second: Since there would be no way to tune the levers after mounting them, extreme and very time consuming care
would be required to be
sure the lever is placed exactly right when it is installed.
Third: There is some risk involved. If there is any significant error in drilling the first mounting hole, very
little can be done to correct that.
Now, I suppose the question is; Do these three reasons justify using "tunable frets" that can and do go out
of tune. That, of course,
depends entirely on the harpmaker and his/her personal business philosophy. Experience has shown that a harpmaker can
reasonably actually apply specifications to the accuracy of his levers if he so chooses. A reasonable such specification
might be within +/- 14 cents of perfect accuracy at the time of delivery, and +/-16 cents of perfect accuracy throughout
the life of the harp. (One cent is 1/100 the interval of one evenly tempered half step.
***For any who may be interested and might like to have a copy of the mathematical derivations found in this article,
I do have them available. In general they do not go beyond a good course in highschool Algebra II. Some small knowledge
of the physics of sound would also be useful in following the reasoning. In the same manner that I previously suggested
you check my statements re the bowing out of the soundboard, I would welcome any critique concerning this work. If you
happen to know a Jr College professor of Physics, or even a HS AP physics teacher could very easily review the
mathematical reasoning.
For fairly good related articles on "intonation and intervals" and their vocabulary, I suggest the
"New Harvard Dictionary of Music" My copy is the 1986 edition, but there may be later and I’m very sure
equally as good editions.
Paul Culotta
Woldsong Harps
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