For this example we are going to reconstruct the stringband length of an Erard concert harp from a beautiful technical drawing that was made in Paris, in 1901,

Why am I using a concert harp design when most of us make or play folk harps? 

1.I have never seen a technical drawing of this quality for a folk harp! 
2.Concert harps had and still have a strong influence on folk harp. 
3.It is a beast that needs to be demystified.
4.The Erard harps are legendary. 
5.But  most of all, it is the perfect example for demonstrating the importance of establishing the theoretical harp curve. This will allow us to reconstruct the stringband although the drawing has no scale or any reference to a system of measurement.

Let us look at the drawing. If you compare the drawing to a picture of an Erard harp you will see that the smallest little details seem to have been drawn. The drawing is clean. It looks so precise. At the bottom of the drawing there is a N.B. that says "This technical drawing has been established from pictures of an Erard Harp." However there is no scale reference to the drawing. The C and F strings are drawn in thicker lines. This helps us to determine the harp range which is C1 flat to G7 sharp. Middle C (C4) is string number 26.

IDENTIFYING THE METHODOLOGY TO FOLLOW:
Here is what we are going to do to reconstruct the full size stringband from the drawing.
1.Measure precisely the stringband from the drawing.
2.Establish a theoretical stringband for the treble part of the harp.
3.Establish the scale ratio of the drawing.
4.Convert all drawing measurements to real life size.
5.See a string configuration example for that harp.

MEASURING THE STRINGBAND FROM THE DRAWING:

First down load the picture and print it as large as you can then take it to a photocopier shop and have the drawing enlarged by 200% or more. We need great accuracy while measuring your initial data because any error you make while measuring the stringband will be multiplied by the scale ratio we will establish latter on. Use a transparent ruler that has hair line type of mark with small graduation units such as the millimetre or 60th of an inch (engineers ruler). Also, use a good magnifying glass to help the positioning of the ruler and the reading. Using this technique I can be precise to about 0.2 millimetre (0.008").

On a concert harp each string can produce three notes (flat, natural, sharp) except for the very top or bottom ones. Measure the vibrating length of all the strings as if there were all in the flat mode because this is where we have the greatest precision. Measure also the spacing and the soundboard length between the first and last string, and the angle of the strings from the sound board. Number the strings from 1 to 47 from the top to bottom. Label each string with its musical note value in the flat mode, from G7 flat to C1 flat. We will use the tempered scale so for my string computer program G7 flat will become F7 sharp and C1 flat will be B0.

ESTABLISHING THE THEORETICAL STRINGBAND FOR FIRST 10 STRINGS.

Usually on a harp stringband there is a string that has the greatest tensile strength ratio. It is most often located in the treble section of the harp, let us say within the first 10 strings. For this harp we will assume that it had gut strings (nylon did not exist in those days) with a tensile strength ratio of 70 percent (concert harps are known to have tight strings). Let us take a 0.025" gut string to fuel our theoretical string's length formula. (See part three of Stringband evaluation).

The first string is G7 flat, frequency is 2960 c/s, From my tables The linear mass for a 0.025" nylon string is M = 5.9629E-08  and tension at 70% is T = 17.9 lb. Therefore Length = square root of (17.9/0.000,000,059629) divided by (2*2960) = 2.92 inches. Do this for the first 10 strings. Set up a spreadsheet with Column A for string number, Column B for note, Column C for the frequency and Column D for the length. The length formula is @sqrt(17.9/0.000000059629)/(2*C1). C1 refers to the frequency cell for that string.

Using my program, choose the length method of calculation. If you want to use a different percentage strength ratio like 60%, you will have to calculate the tension of the string when tuned at 60% of its breaking point. Lowering the tensile strength ratio will also shorten the theoretical length of the string.

ESTABLISHING THE SCALE RATIO OF THE DRAWING: 

Dividing the theoretical length by the string length from the drawing will give you a ratio. One string will have the smallest ratio. That string is the string we will be looking for. It has the highest tensile strength ratio on the harp (we decided it was 70%, it could be whatever you want). If that string is close to string #10, do more strings to make sure that none other has a smaller ratio number. Here is what I found from my drawing:

 
ST#Note Freq     DrawL TheoL  Ratio
1G7 flat 2960.0  0.472 2.922  6.19
2F7 flat 2637.0  0.551 3.280  5.95
3E7 flat 2489.0  0.629 3.475  5.52
4D7 flat 2217.5  0.708 3.900  5.51
5C7 flat 1975.5  0.787 4.378  5.56
6B6 flat 1864.7  0.866 4.638  5.35
7A6 flat 1661.2  0.944 5.207  5.51
8G6 flat 1480.0  1.023 5.844  5.71
9F6 flat 1318.5  1.141 6.560  5.75
10E6flat 1244.5  1.259 6.950  5.52

String number 6 is the winner and the scale ratio for my drawing is 5.35. Note that your number will be different from mine because it is very unlikely that you will measure the same string length from your drawing. For this example, I measured the lengths in millimetres then converted them into inches by using the 0.03937 conversion factor.

CONVERTING ALL DRAWING MEASUREMENTS TO REAL LIFE SIZE.

This is easy. Multiply your drawing measurements by the scale ratio you have.You will have to smoth the curve line of the string band of the neck because of measurement accuracy.  For me, the ratio is 5.35. This gives me a length of 63.189 inches for the last string. A spacing of 28.224 inches between the first and the last string. The length of the sound board between the first and last string is 50.972 inches. The strings are at 34 degrees from the sound board. The life size measurements may not be exactly the same as a real Erard Harp but we have captured its spirit.

STRING CONFIGURATION FOR THIS ERARD HARP:

The only help I was able to find to guide me with string type is from the book "How to play the harp" by Melville Clark. It says that concert harps have compound wire strings starting at E2 (string # 36). Let us see what we end up with when using gut strings up to string #35 and compound wire strings from there after. This is just a stringing example for that harp based on the parameters we have stipulated above. 

One can use the stringband we have just reconstructed to design a folk harp by adjusting the stringband to host nylon strings instead of gut. We can easily find the string length for the natural note (as opposed to the flat) of each string by multiplying the length by a decreasing factor of 5.612 percent. (Use global edit from program). If you delete the first 4 strings and the last 7 strings you have a 36 nylon strings harp (C2 to C7) that you may claim has an Erard flavour. What you now have is a sound stringband length based on a historical harp. You can apply this technique to any good drawings or pictures of a harp of your choice. The "Harps & Harpists" book by Roslyn Rensch is an excellent source for good historical harp pictures.

What I wanted to demonstrate in this article, (besides empowering harp makers with new approaches to harp design), is how powerful and useful our mathematical tools can be in enhancing a research, design or "fixing up" job. In this case it was with the use of the theoretical string's length formula which can be easily used even if you don't have a computer. I believe this is the first step toward a good understanding of stringband design.

This example also illustrates how important it is to have a good source of historical or contemporary harp stringbands.

 Eard Harp tutorial: units in inches and pounds. Stringing example variation

STR#	NOTE	FREQ	LENGTH	CMAT	WMAT	CDIA	WDIA	BDIA	ODIA	TENSION
1	g7	3136.000	2.386	Nylon		0.025	0.000	0.000	0.025	10.976
2	f7	2793.800	2.783	Nylon		0.025	0.000	0.000	0.025	11.851
3	e7	2637.000	3.181	Nylon		0.025	0.000	0.000	0.025	13.794
4	d7	2349.300	3.579	Nylon		0.025	0.000	0.000	0.025	13.859
5	c7	2093.000	3.976	Nylon		0.025	0.000	0.000	0.025	13.576
6	b6	1975.500	4.374	Nylon		0.025	0.000	0.000	0.025	14.637
7	a6	1760.000	4.771	Nylon		0.025	0.000	0.000	0.025	13.822
8	g6	1568.000	5.169	Nylon		0.028	0.000	0.000	0.028	16.154
9	f6	1396.900	5.765	Nylon		0.028	0.000	0.000	0.028	15.948
10	e6	1318.500	6.362	Nylon		0.028	0.000	0.000	0.028	17.303
11	d6	1174.700	6.958	Nylon		0.030	0.000	0.000	0.030	18.859
12	c6	1046.500	7.555	Nylon		0.032	0.000	0.000	0.032	20.077
13	b5	987.770	8.151	Nylon		0.032	0.000	0.000	0.032	20.821
14	a5	880.000	8.748	Nylon		0.032	0.000	0.000	0.032	19.034
15	g5	783.990	9.344	Nylon		0.036	0.000	0.000	0.036	21.815
16	f5	698.460	10.338	Nylon		0.036	0.000	0.000	0.036	21.194
17	e5	659.260	10.934	Nylon		0.036	0.000	0.000	0.036	21.122
18	d5	587.330	11.929	Nylon		0.036	0.000	0.000	0.036	19.954
19	c5	523.250	12.923	Nylon		0.040	0.000	0.000	0.040	22.947
20	b4	493.880	13.917	Nylon		0.040	0.000	0.000	0.040	23.709
21	a4	440.000	15.109	Nylon		0.040	0.000	0.000	0.040	22.180
22	g4	392.000	16.302	Nylon		0.045	0.000	0.000	0.045	25.938
23	f4	349.230	17.694	Nylon		0.045	0.000	0.000	0.045	24.253
24	e4	329.630	19.086	Nylon		0.045	0.000	0.000	0.045	25.140
25	d4	293.660	20.676	Nylon		0.050	0.000	0.000	0.050	28.908
26	c4	261.630	22.465	Nylon		0.050	0.000	0.000	0.050	27.089
27	b3	246.940	24.454	Nylon		0.050	0.000	0.000	0.050	28.594
28	a3	220.000	26.243	Nylon	Nylon	0.045	0.008	0.000	0.061	35.098
29	g3	196.000	28.827	Nylon	Nylon	0.045	0.008	0.000	0.061	33.614
30	f3	174.610	31.213	Nylon	Nylon	0.050	0.008	0.000	0.066	36.873
31	e3	164.810	33.996	Nylon	Nylon	0.050	0.008	0.000	0.066	38.969
32	d3	146.830	36.382	Nylon	Nylon	0.050	0.013	0.000	0.076	45.407
33	c3	130.810	38.967	Nylon	Nylon	0.055	0.013	0.000	0.081	47.285
34	b2	123.470	41.551	Nylon	Nylon	0.055	0.013	0.000	0.081	47.900
35	a2	110.000	43.738	Nylon	Nylon	0.060	0.013	0.000	0.086	47.783
36	g2	97.990	46.720	Nylon	Nylon	0.060	0.016	0.000	0.092	48.780
37	f2	87.310	48.112	Nylon	Nylon	0.060	0.020	0.000	0.100	47.745
38	e2	82.410	48.907	Nylon	Nylon	0.060	0.022	0.000	0.104	47.219
39	d2	73.420	50.497	Steel	Bronze	0.020	0.008	0.010	0.046	48.835
40	c2	65.410	51.889	Steel	Bronze	0.020	0.010	0.010	0.050	49.930
41	b1	61.740	53.082	Steel	Bronze	0.020	0.010	0.014	0.054	50.873
42	a1	55.000	54.275	Steel	Bronze	0.022	0.013	0.009	0.057	53.218
43	g1	49.000	55.468	Steel	Bronze	0.022	0.013	0.016	0.064	50.823
44	f1	43.650	56.661	Steel	Bronze	0.024	0.013	0.021	0.071	48.861
45	e1	41.200	57.655	Steel	Bronze	0.024	0.013	0.026	0.076	49.053
46	d1	36.710	58.649	Steel	Bronze	0.026	0.016	0.025	0.083	50.985
47	c1	32.700	59.643	Steel	Bronze	0.026	0.020	0.025	0.091	52.693

Data Graph of  Erard Harp tutorial.