String tension in silk and steel/nylon strings

By Dan Nung Ing, 1st May, 2005

Introduction

I have been to China four times during the last six months. During that time I met Professor Chen Changlin (professor of computer science at the Academy of Science in Beijing and an eminent qin player), who has been studying the tension of silk and steel/nylon qin strings. He remarked that there is now a tendency among the new generation of qin makers and players to wrap strings 1-3 around one of the "goose feet", and 4-7 around the other. Traditionally, however, strings 1-4 are wrapped around one foot, and strings 5-7 around the other. Why are they breaking with thousands of years of tradition?

Method

It might look better to have the thickest 3 and the thinnest 4 strings grouped together, but we need to consider not their appearance but their tension and musical quality. So I did some calculations to determine the tensions of both types of strings. There are two methods by which this may be done: directly, by tensioning the strings with weights at one end, or indirectly, by calculation. With the direct method, you add weights to one end of each string until the desired pitch is obtained. The calculation method uses a formula, well known to students of physics, which relates the length, tension, mass and frequency of a vibrating string. With the calculation method, the first requirement was the precise frequency for each open string. These were provided by Christopher Evans.

I prepared two sets of used strings, one of Suzhou silk and one of Shanghai Conservatory steel/nylon. I cut off the butterfly knots. For silk strings 1-4, I also cut off the part which has no outer wrapping at each end, leaving pieces approximately 1.5 in length. For silk strings 5-7, there is no outer wrapping along any part of its length, so I had 2m long pieces. This preparation ensured that the pieces were of uniform construction. In steel/nylon strings, both the butterfly knots and the part which wraps around the goose feet, about 2 feet in length, are also of a different construction to the vibrating part, so I cut them off too. I weighed all these pieces of string on an electronic balance accurate to 0.001g at Imperial College, London University. Both silk and steel/nylon strings weighed between 2g and 5g, depending on their thickness and length. The length of the vibrating part of each string was 1.121 m.

Note: There is very little difference in the length between the outermost and the middle strings, e.g. between the 1st and 4th strings. However, in theory at least, the outermost strings should be slightly longer than those in the middle.

The formula used to calculate string tensions was:

string tension = 4 x (string length x frequency)² x density

Units used were:

Tension: Newton (1 kg = 9.98 Newton)

String length: metre

Frequency: hertz

Density: kg per cubic metre

The estimated error in the tension calculation was about 5-10%.

Results

For silk strings, the tensions were:

String	S1	S2	S3	S4	S5	S6	S7
Frequency (Hz)	64	72	85.33	96	108	128	144
Relative tension	1	1.07	1.36	1.50	1.41	1.36	1.36
Absolute tension (Newtons)	57.7	61.6	78.1	85.9	80.9	77.9	78.2

Total tension for strings S1-S3: 197.4 Newtons
Total tension for strings S4-S7: 322.9 Newtons

Total tension for strings S1-S4: 283.3 Newtons
Total tension for strings S5-S7: 237.2 Newtons

Total tension for all 7 strings: 520 Newtons

For steel/nylon strings, the tensions were:

String	S1	S2	S3	S4	S5	S6	S7
Frequency (Hz)	64	72	85.33	96	108	128	144
Relative tension	1	1.12	0.79	0.82	0.90	0.84	0.86
Absolute tension (Newtons)	83.1	93.0	65.7	68.4	74.6	69.5	71.3

Total tension for strings S1-S3: 241.8 Newtons
Total tension for strings S4-S7: 283.8 Newtons

Total tension for strings S1-S4: 283.3 Newtons
Total tension for strings S5-S7: 237.2 Newtons

Total tension for all 7 strings: 525 Newtons

Conclusions

The difference between the combined tensions of all 7 strings for silk and steel/nylon strings was less than 1%.

There is of course variation between strings of different makes, and the study should be repeated using a variety of different makes.

For silk, based on adding together the tensions of the strings, it makes sense to group strings 1-4 together. But there is a much more important reason to do this: strings 4 and 7 are the ones that break the most often. The inner core of a silk string 4 is as thin as that of string 7. Since these strings break so often, they should be the outermost strings for easy replacement without disturbing the other strings. Steel nylon strings do not have an outer wrapping, and string 4 does not break so easily, so it does not matter how they are grouped. The total tension for all 7 strings is about half that of a guitar. In terms of stress and strain, it doesn't matter how you group steel/nylon strings. However, Wang Peng remarked that strings 1-4 should be grouped together because otherwise the bass of string 4 might be reduced.

This study was presented at a London Youlan Qin Society yaji on 1st May, 2005. For the report on that yaji, please click here.