String Band analysis from historical drawing of an early Cousineau harp

© Joseph Jourdain – Josephus Harp Shop – 2019

 

To illustrate how we can use Stringband Manager V4.2 to design a new harp from an old drawing we will investigate the single action pedal harp made by Coussineau, in France, in the late 18th century. (It can also be done with my freeware calculator SBM LIGHT)

 

 

Download large jpg picture of drawing here: cousineau-big 12.4MB

Here is a copy of a harp drawing that dates from the 18th century. Originaly from "L'encyclopédie Diderot et D'Alembert - Lutherie". We will investigate that drawing (Cousineau single action harp). We will try to figure out how it was strung. As we will see, there will be a problem when it is time to string that instrument. That is why I choose this example. It will demonstrate how we can change a harp design in order to accommodate the string material we are using nowadays and the norms used in modern folk harp making. The investigation will also tell us what strings they must have had in those days in order to make the harp playable.

You will notice on the drawing the letters "ut" on some of the strings. This is an Italian word meaning "Doh" or the "C" notes in English. From this we can determine that Middle C is string number 20. When working at a complex project such as this it is a good practice to lay down the major steps of your methodology. If you get weird results it will be easier to go backwards and see where you made a mistake. If you do everything on the fly you may not be able to check your results with confidence. Here is the methodology we will use to investigate the Coussineau harp.

1) - Establishing the scale ratio
2) - From the drawing's scale we will establish the stringband length
3) - Establish the way the harp was tuned. (Flat or natural mode)
4) - Establish the strength of the gut they must have been using
5) - Find the theoretical length one must use for various string materials
6) - See a nylon stringband configuration of that "new" harp

1) - ESTABLISHING THE SCALE RATIO:

First let us look at the drawing. The scale for the drawing refers only for the harp and must not be used for the parts that are drawn around it. Also the scale length refers to the French foot and is not the same as the British foot. We need to establish the scale ratio for that scale. For this we divide the unit length by the length measure on the drawing provided that both measurements use the same unit system. One French foot equals 324.84 millimetres or 1.0657 British foot. In this example the scale ratio is equal to a French foot (1 X 1.0657) divided by the length of the "Echelle" (scale) you measured (in feet) on your enlarged copy of the drawing (you can convert feet into inches if you prefer). That number is the scaling ratio you will use to determine the real size of the string band. You should find that the longest string is about 59 inches long and the smallest 4.866 inches.

2) - MEASURING AND ESTABLISHING THE STRINGBAND FROM THE DRAWING:

Down load the picture and print it as big as you can. Take your print to a good 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 you have established.  Use a transparent ruler that has hair line type of markings with small graduation units such as the millimetre or 60th of an inch (engineers ruler). Also, use a good magnifying glass to help with the positioning of the ruler and the readings. Using this technique I can be precise to about 0.2 millimetre (0.008"). Measure the vibrating length of all the strings as if all levers were opened 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 34 from the top to bottom. Label each string with its musical note value. Multiply each value by the scale ratio you have determined earlier using the same unit of measurement. There you have the initial data for your investigation

3) - ESTABLISHING HOW THE HARP WAS TUNED

Since this is a single action pedal harp, the harp can be tuned either in a flat or natural mode when the levers are not on. Our first assumption is that A4 = 440 c/s although many Baroque instruments were known to have been tuned with A4 = 415 c/s. How can we find this out? We will calculate the tensile strength ratio of the first 12 strings in the natural mode and see how far they are from their breaking point. If the highest tensile strength ratio is not too tight we will assume the strings were tuned in the natural mode, if it is too tight we will tuned the string in the flat mode and if it is still too tight we will tune the string in the flat mode based on Baroque tuning A4 = 415 c/s.

Using our data and SBM4 we will find that string number 7 has a length of 10.79 inches and that tuned at B5 (even tempered scale) with a 0.028 inch gut string it will have a tension of 34.034 lb that will give a tensile strength ratio of 106.3 percent. That ratio is well above the expected level. We must conclude that the harp was tuned in the flat mode. Tuning that same string to B5 flat we will find that it has a tension of 30.3 lb and a tensile strength ratio of 94.7 percent. This is still too high a percentage. Let us try the Baroque tuning with B5 flat = 879.3 c/s (use A5=880 c/s). This gives a tension of 27.012 lb and a tensile ratio of 84.3%. This string will break after a few tunings. What is going on? They must have used a gut string that is stronger that the one I used here (52000 Lb per square inch).

4) - FINDING THE STRENGTH OF THE GUT THEY MUST HAVE BEEN USING.

We have determined that the harp was probably tuned in flat mode with a Baroque tuning based on A4 = 415 c/s. From our already collected data we can easily determine the strength of the gut they must have been using. If my 0.028" gut string breaks at 32.019 Lb, then the original gut string should have broken at 38.6 Lb in order to be at 70% of its tensile strength for a pull of 27.012 lb. From this we find that the tensile strength for a squared inch of that old string must have been 62687 lb. This is quite a difference from my 52000 lb. A detailed study of the history and technology of plucked string instruments by Franz Jahnel (1981) includes the information that the major string gut manufacturers make 3 grades of gut string. Grade 1 ÷ 65000lb/in², grade 2 ÷ 55000lb/in² and grade 3 ÷ 45000lb/in². If we use the grade 1 gut string we will find that the string #7 has now a tensile strength ratio of 67.5 percent which is quite acceptable. We have resolved our mystery. However, if we want to use this stringband for the basic design of a new harp we have to make some modifications in order to fit modern pitching (A=440) and the string material that you have access to. Let us therefore modify the overall stringband length.

5) - FINDING THEORETICAL LENGTH FOR GRADE #2 GUT AND NYLON STRING

We will calculated the theoretical length for string # 7, tuned at B5 flat  even tempered scale, using a maximum tensile strength of 70 percent and for grade #2 gut (commonly available) and nylon strings. For grade 2 gut we find that the string should be 9.77 " long, which is 1.020"  shorter than our original length and for nylon the length should be 9.483" long or 1.307" shorter. Now you can use that knowledge to modify your entire stringband. If you choose to make the grade 2 gut version of that harp you will shorten all the strings by 1.020" and for a nylon version you will shorten all the strings by 1.307". This is not all, for remember that we calculated our theoretical length based on a flat tuning of the note. Most folk harps however are tuned in the natural mode so we need to shorten our new stringband by a decreasing factor of 5.612 percent. Use the Global Edit option for both of these operations. If you reduce the length all the strings by the same amount you are lowering the position of the neck while keeping the original neck curve and its relationship to the soundboard.

Here you have it, a new harp design based on an original  Coussineau stringband. What you need now is to establish the string spacing and the string angle from the sound board. I found that the string angle from the sound board is about 28 degrees and that the total spacing over all the strings is about 19.35 inches. You are all set up now. Draw your "new harp" according to your results.

Notice that the Coussineau Harp has more of a folk harp flavour that a traditional concert harp. The neck is not as curved and the string angle is much smaller. The middle and bottom strings are much longer than our modern concert and folk harps for comparative string's pitch. This suggests to me that they probably did not use composite strings for the bass section. Those are distinct features that are unique to it. I wonder how it will sound now! Enclosed is a stringband configuration for such a harp tuned in natural mode and using nylon strings. The string lengths have been arrived by using the method I have demonstrated in this paper. With SBM V4 it took me only a few minutes to do that kind of stringband evaluation. In the table below you have the original data and the modiflied version using even tempered tuning and nylon stringing (you could choose gut if you wish). Note that for the nylon version the first three strings have been lengthend to be more in unison with the following ones. More modifications could be done but as you modify only specific areas of the stringband you are moving away from the original design.

I hope you find these stringband strategies useful for your work, that it will give you new ideas in harp designing and that I have demonstrated how easy it is to copy a stringband design from a little drawing. You can do the same from a good picture for research. I also hope that I have demonstrated how important it is to work out your stringband before you start making a harp. Take nothing for granted; do your calculations.

Cousineau analysis tutorial for a new harp: units are inches and pounds

Original

Original

Original

Modified

Modified

Modified

 

 

STR#

NOTE

FREQ

LENGTH

NOTE

FREQ

LENGTH

CMAT

WMAT

CDIA

WDIA

ODIA

TENSION

1

g#6

1661.200

4.866

a6

1760.000

4.371

Nylon

0.028

0.000

0.028

14.555

2

f#6

1480.000

5.924

g6

1568.000

4.953

Nylon

0.028

0.000

0.028

14.830

3

e6

1318.500

6.982

f6

1396.900

5.625

Nylon

0.028

0.000

0.028

15.183

4

d#6

1244.500

7.828

e6

1318.500

6.155

Nylon

0.028

0.000

0.028

16.196

5

c#6

1108.700

8.674

d6

1174.700

6.954

Nylon

0.028

0.000

0.028

16.408

6

b5

987.770

9.732

c6

1046.500

7.952

Nylon

0.028

0.000

0.028

17.030

7

a#5

932.330

10.790

b5

987.770

8.951

Nylon

0.028

0.000

0.028

19.223

8

g#5

830.610

11.847

a5

880.000

9.948

Nylon

0.028

0.000

0.028

18.848

9

f#5

739.990

12.905

g5

783.990

10.947

Nylon

0.032

0.000

0.032

23.658

10

e5

659.260

13.963

f5

698.460

11.946

Nylon

0.032

0.000

0.032

22.360

11

d#5

622.250

15.232

e5

659.260

13.144

Nylon

0.032

0.000

0.032

24.115

12

c#5

554.370

16.608

d5

587.330

14.442

Nylon

0.032

0.000

0.032

23.110

13

b4

493.880

18.089

c5

523.250

15.840

Nylon

0.036

0.000

0.036

27.926

14

a#4

466.160

19.570

b4

493.880

17.238

Nylon

0.036

0.000

0.036

29.463

15

g#4

415.300

21.156

a4

440.000

18.735

Nylon

0.036

0.000

0.036

27.624

16

f#4

369.990

22.743

g4

392.000

20.233

Nylon

0.040

0.000

0.040

31.570

17

e4

329.630

24.330

f4

349.230

21.731

Nylon

0.040

0.000

0.040

28.904

18

d#4

311.130

26.445

e4

329.630

23.727

Nylon

0.040

0.000

0.040

30.699

19

c#4

277.180

28.561

d4

293.660

25.725

Nylon

0.045

0.000

0.045

36.247

20

b3

246.940

30.677

c4

261.630

27.722

Nylon

0.045

0.000

0.045

33.412

21

a#3

233.080

32.792

b3

246.940

29.718

Nylon

0.045

0.000

0.045

34.206

22

g#3

207.650

35.437

a3

220.000

32.215

Nylon

0.050

0.000

0.050

39.387

23

f#3

185.000

38.081

g3

196.000

34.710

Nylon

0.050

0.000

0.050

36.293

24

e3

164.810

40.726

f3

174.610

37.207

Nylon

0.055

0.000

0.055

40.047

25

d#3

155.560

43.370

e3

164.810

39.702

Nylon

0.055

0.000

0.055

40.624

26

c#3

138.590

46.015

d3

146.830

42.199

Nylon

0.060

0.000

0.060

43.351

27

b2

123.470

47.919

c3

130.810

43.996

Nylon

Nylon

0.050

0.008

0.066

41.116

28

a#2

116.540

49.717

b2

123.470

45.693

Nylon

Nylon

0.050

0.010

0.070

43.777

29

g#2

103.830

51.515

a2

110.000

47.390

Nylon

Nylon

0.050

0.013

0.076

43.240

30

f#2

92.500

52.891

g2

97.990

48.689

Nylon

Nylon

0.055

0.016

0.087

47.079

31

e2

82.410

54.477

f2

87.310

50.186

Nylon

Nylon

0.055

0.020

0.095

46.596

32

d#2

77.780

56.064

e2

82.410

51.684

Nylon

Nylon

0.060

0.020

0.100

49.087

33

c#2

69.300

57.651

d2

73.420

53.182

Nylon

Nylon

0.060

0.025

0.110

49.140

34

b1

61.740

59.000

c2

65.410

54.480

Nylon

Nylon

0.060

0.030

0.120

48.125

Data Graph

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