Question for harp builders??

[carl-swanson]

There is a formula for the percentage that a string has to be shortened to raise the pitch a half step. Does anyone know what that formula is? I may not be stating this correctly. Put another way: What is the percentage of any string length that the string needs to be shortened to raise the pitch a half step? I used to have it. But I can’t find it, and I don’t remember where I got it.

[carl-swanson]

I seem to remember that the percentage number was something like 6.something,something,something. But I just don’t remember.

[charles-nix]

Carl, assuming you’re saying “reduce the length at a constant diameter and density”, which is what you would need for mounting levers, but not for calculating a whole stringing, divide the length of the longer string by 1.059463094 1.0595 ought to be close enough. If you want to multiply, the number would be 0.943874313 or about 0.94387.

The number is the 12th root of 2, because if you take a string length, and multiply it by the 12th root of 2 twelve times (one for each semitone) you arrive at a length exactly twice the original length, and a pitch exactly one octave lower. 2^(1/12) = 1.059463094.

Does that help?

Organbuilders would express that as “12th halving” because the number halves every 12 notes.

[Biagio]

Charles has given the exact derivation (yay Charles) which would result in a percentage shortening of 5.613%..but as a practical matter, if mounting levers I just estimate the sharp point as 0.056xmeasured length. Test and adjust as needed. Published lengths never match the actual ones for one thing and it is easy to see how even one millimeter off could make a huge difference as one approaches the higher notes.

[carl-swanson]

I knew I could count on you two!! Thank you both. I’m going to have to print this out and get my partner, who is an MIT graduate, to explain it to me. I’m a total dunce for these kinds of things.

I actually didn’t ask for lever harp installation. I’m going to have to do some major repair work on a pedal harp make that was often inconsistent in critical dimensions. Over the years I’ve regulated them and occasionally found that whole sections of the instrument were un-regulateable, especially the wire strings. Often, that whole area was way flat, and there was nothing I could do to fix it. I’m putting a new soundboard on this harp, and I want some formula to try to figure out what the string lengths should be down there so that the half steps are at least in the ball park. I can alter the height of the column ring if I have to to make corrections. But I want some kind of formula to show what those lengths should be. I’m also going to take some measurements off several harps that I have where the lengths are correct, and use those measurements to compare this instrument to.

I know that the formulas you put forth are correct in theory, but may not be in actual practice, at least not over the whole instrument. They are probably most correct in the lower strings. Down there, the string lengths are so long, and the added tension of the closed disc so insignificant that the formulas are probably correct. But as you go higher on the instrument, the tension added by the closed disc has more impact, and so the distances calculated by these formulas would have to be modified to accommodate that. But thank you both. This gives me something to start with.

Here is the same question from a different standpoint: Let’s say I’m trying to calculate the string lengths needed from 5th octave C to 7th octave E, which is all the strings with stationary nuts (i.e., not adjustable at all). If I know the distance from the point on the stationary nut where the string touches (i.e.flat position) to the lower prong of the natural disc, how do I use that number to calculate what the full string length should be? I could then measure the distance from the lower prong of the natural disc(i.e., natural position) to the lower prong of the sharp disc, and do the same calculation, which would tell me what the string length should be from natural position to the soundboard. If that calculation did not jive exactly with the measurement from flat position, I would know that I have to slightly compromise on the string length to get natural and sharp positions at least in the ball park.

[Gregg Bailey]

I was going to ask if there’s some reason that the answer to the original question isn’t simply the string length multiplied by the 12th root of 2, but, of course it wouldn’t be that simple! As Carl mentioned, however, maybe that’s a good starting point, especially in the longer strings.

[charles-nix]

I’ll take a stab at the algebra—

Let k = 12th root of 2, let l = total length from nut to soundboard, let f = length from nut to lower prong of natural disk.

f = l – [length from lower prong to soundboard]
or
f = l-(l/k)

multiply every term by k gives:
kf = kl -l = (k-1)l

divide both sides by (k-1):
kf / (k-1) = l

(in words): multiply the length from nut to lower prong of natural by k (12th root of 2), then divide by k-1, giving the total string length nut to soundboard.

The same equation can work for natural to sharp: just use the lower prong natural to soundboard length for the nut to board length, and the natural to sharp prong length is what is measured.

It will take careful measurement. Any error is multiplied by about 20 times in the whole string length.

Carl, you (and Biagio, and others) know this, but for anyone who may search and find this in the future, remember that the length to the soundboard is the length _after_ the soundboard has “bellied” up. And as you point out, Carl, the prong contact points need to leave the pitch slightly flat, so that the increased string tension from their grip will pull the string on up to pitch.

Maybe you have an easier method by comparing with a harp that will regulate. Measure the ratios of lengths (nut to natural, natural to sharp, sharp to soundboard) on several notes of a harp that regulates well, then keep those ratios the same on the harp you’re modifying. The “good” harp will have already accounted for soundboard pull-up and disc grip in each register of strings, so there is no guessing about that.

[carl-swanson]

Charles- Thanks for the math! After my last post, I thought of perhaps another way of approaching this. How about calculating the total string length off the measured distance from flat to natural, as I wrote in the post. But then, instead of calculating the string length from natural to the soundboard, calculate the total string length based on the distance from flat to sharp. In other words, calculate the total string length based on two measurements: the action’s distance from flat to natural, and it’s distance from flat to sharp. In both calculations the final numbers should be the same. If they are not, or if they are off slightly, that would tell me that the distance between natural and sharp position on the action is off. So for that kind of calculation, what are the numbers I need to use to calculate off a whole step distance(flat to sharp)?

Your statement about a soundboard that is “bellied up” is correct, in theory. In practice, I’ve usually found that, even on a badly “bellied up” soundboard, the longest strings, i.e., the wires, are rarely very far off. So there is I believe a fair amount of leeway on the overall string length where it just won’t make that much difference. I’m going to venture a guess that the overall length of a wire string could be off by as much as 1/8 inch (and possibly more!) and not significantly effect the pitch. So on an instrument like I originally mentioned, where the wires were almost 1/4 step flat, the string lengths must be off by a lot.

This is reinforced by the fact that, when I regulate a harp, changing a disc to a larger or smaller disc to correct the regulation, especially in sharp position, is really only effective, at most, in the top 3 octaves. Below that, the string lengths are simply too long to be able to alter the pitch with a larger or smaller disc.

What I would like to do at this point is take some measurements (flat to natural, natural to sharp, and flat to the soundboard) of the wire strings on 3 or 4 of my 23’s, on which the wires are well regulated. Then I want to do some calculations on those same distances and see what the calculation says the length ought to be. In other words, how much of a difference is there between the theoretical length, and the actual length?

[charles-nix]

Measuring and comparing sounds like a good idea. The hard part is measuring accurately enough.

Wow–1/4 step flat! I was thinking you were talking 8-10 cents when you said it was really off. No wonder you want to find a way to fix it!

Think about the math for two semi-tones this way: each semi-tone divides the starting length by 12th root of 2, which I called “k” above for easy reference. So two semi-tones divides starting length by k, and that result divided by k again. Or, length divided by k^2 (squared), which is also 6th root of 2, which is also the square root of the cube root of 2.

In the formula you referred to, you can use k^2 in place of k everywhere to calculate for whole tone lengths. 6th root of 2 is 1.122462048. 1.1225 should be plenty close enough. It will be hard to measure even to 4 significant digits (that would take measuring accurately to 1/128″ in fractions or 1/5mm in metric), so the inaccuracy in measuring will be the largest error.

I was thinking on how to measure nut to prong or prong to prong accurately enough. Best I’ve thought of is using a depth mic, if you’ve one long enough. Should get pretty close with a dial caliper also, though. That will get you readings in thousandths, which will work in the math a lot easier than handling the fractions from a ruler. Working all in metric would be even easier if you’ve a tape or rule long enough for the strings.

Take care with Henri coming.

[paul-knoke]

Sam Pratt said the distance for a half step was 1/18 of the sounding string length. A critical point is to make the calculation for the 1/2 step from flat to natural, using the flat string length, then calculate again from natural to sharp using the natural string length. I have a Schimmeyer harp, which has some interesting features, but he used the same distance for natural to sharp as for flat to natural, so a lot of the instrument is useless in sharps – way too sharp.
Hope I explained this in a way that makes sense?

[billooms]

As a former (recovering) engineer, I understand the math. But don’t the disks (or levers) will also increase the tension on the string a bit causing the string to go sharp a bit more than the formula would predict?

[charles-nix]

Yes. Carl, Biagio and I have all touched on that, but all the mentions are buried in the middle of some long posts.

Reading back, several things were left unsaid, because we knew that Carl is one of the finest harp technicians in the US, and a former pedal harp manufacturer. He also wrote The Book on harp maintenance for harpists. We knew he was just asking for the math parts.

[charles-nix]

As a recovering engineer, how would you approach measuring from where the string is tangent to the groove in the stationary nut to where it touches the lower disc prong?

[balfour-knight]

This is a great thread, guys! I’m glad you all know what you are talking about. My sister has a Master’s Degree in math, and she helped me figure out all the formulas when I built my pipe organ. I agree with Carl, I know “very little” about all the math, ha, ha!

Best wishes to you all,
Balfour

[carl-swanson]

I took some measurements from a 23 that I have, measuring some of the wire strings the way I described above. I’ll post those measurements here probably tomorrow (I’m too tired now to deal with numbers!). I’ll be curious to see what you guys can come up with for the overall length of the string, based on the measurements I give you. I’m not going to tell you what the full length of the string is. I want to see what you come up with.

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