1

Tuning Stabilisationfor the Harp

Nicolas Lynch-Aird

Presented at HarpFest 2012, Saint Columba’s House, Woking, UK; organised by the London and South-East Branch of the Clarsach Society.

( harpfest2012.wordpress.com )

This presentation describes an ongoing research project investigating how a harp

may be kept in tune as the temperature and humidity varies, even while the

instrument is being played.

Preliminary results are presented and a variety of possible tuning stabilisation

approaches are discussed, together with practical considerations for their

implementation.

2

Processor

X

Motors

Pickups

Components for Automated Tuning

A fully automated tuning system requires a number of components:

1) Motorised string winders to make the tuning adjustments.

2) Individual pickups to measure the frequency of each string.

3) A processor and drive electronics to analyse the response of each string; calculate the tuning

adjustment required; and drive the corresponding motorised winder to make the adjustment.

Typically the harp would be switched into a specific ‘tuning’ mode and the strings gently strummed

while the tuning system made adjustments.

This approach has already been developed for the Gibson Robot Guitar:

www.gibson.com/robotguitar

Such an approach should be feasible for the harp, but it would be expensive: a fully automated

tuning solution would add several thousand £’s to the cost of a harp.

More fundamentally the problem is not tuning the harp, but keeping it in tune while it is being

played. The strings sometimes go out of tune so quickly that re-tuning between pieces may

not be enough.

It would be difficult to operate an automated tuning system while the instrument was being played,

because the strings may be fully or partially stopped by the player as well as by the semitone

pedals or levers.

Worse, the open string layout means that a particular string may not be played until near the end

of a piece, by which time it is already noticeably out of tune.

The main drivers of this project are therefore to explore how the strings behave with changes in

temperature and humidity; whether this knowledge can be used to make pre-emptive tuning

adjustments while the harp is being played; and whether a less expensive solution might be

feasible.

3

There is very little published data available on how gut and nylon instrument strings vary with

changes in temperature and humidity.

To investigate this behaviour it has therefore been necessary to construct a test rig specifically for

this purpose.

On the left is a motorised winding shaft and a horizontal bridge pin. This can be used to make

precisely controlled adjustments to the length of the string.

On the right the string is attached to a loadcell for measuring the string tension.

Also on the right is a vertical bridge pin – the two bridge pins are perpendicular to each other. With

parallel bridge pins the string frequency varies slightly with the direction of plucking due to very

small changes in the effective string length between the bridge pins. Mounting the bridge pins

perpendicular to each other avoids this.

In the middle is a motorised plucker which uses a guitar plectrum. Yes, it’s Lego.

Also in the middle is a microphone for recording the pluck responses and a temperature and

humidity sensor.

The black trough surrounding the string is aluminium and its purpose is to even out the

temperature along the length of the string.

The baseboard is a piece of 80 year old Canadian rock maple nearly 7 cm thick.

4

The whole rig is contained within a closed box with incandescent light bulbs to provide heating

and fans for air circulation.

More recently a 5 litre water bottle with its top cut off, filled with water, has been added to keep the

humidity up and prevent the rig drying out excessively.

The fans help to ensure a fairly uniform temperature and humidity throughout the chamber.

5

The whole rig is computer controlled and highly automated.

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No Adjustment - Frequency Deviation vs. Temperature

This slide shows how the tuning of a typical nylon harp string changes with temperature.

The string used is a Nylon 3rd Octave A (A4) Concert Harp string from Bowbrand.

No tuning adjustments have been made. The string has simply been heated up and cooled down

a number of times and its frequency recorded.

The graph shows how the frequency changes with temperature.

The frequency change (deviation) is measured in cents. There are 100 cents in a semitone, 1200

in a full octave. It is generally agreed that the smallest audible change in pitch is 2 cents.

All temperatures are in degrees Celcius (aka centigrade).

As the temperature goes up the string is going sharp at a rate of 2 cents per degree Celcius

(¢/°C). So a 5 °C change in temperature would result in a 10 ¢ change in pitch.

This test was run for multiple heating/cooling cycles – each zigzag on the plot corresponds to one

cycle; starting on the left and slowly progressing to the right.

This means that overall the string is also going slightly flat over successive heating/cooling cycles.

The string frequency is determined by its length, tension, and mass.

The length, between the two bridge pins, is taken to be constant – this is why a piece of well-

seasoned hardwood was used as the baseboard.

7

Variation in Natural String Length with Temperature

Initial Tuning Thermal Expansion Thermal Contraction

Total length

fixed by frame

dimensions and position

of bridge pins

Natural length

Stretching due to Tension

Natural length

increasesNatural length

decreases

Less stretching

Lower Tension

More stretching

Higher Tension

The tension in the string is due to the string being stretched. The more the string is stretched, the

higher the tension and the higher the string frequency. This is how the strings are tuned.

For a given string, the tension depends on two factors: how difficult it is to stretch the string, and

how far it is stretched – the difference between its stretched and natural (unstretched) lengths.

Typically this difference is only a few millimetres; between about 1% and 4% of the total

(stretched) length.

We can measure how difficult it is to stretch the string and it turns out that it actually gets easier as

the string gets warmer. This would tend to reduce the string tension and make the string go flat –

but the string actually goes sharp.

Most materials expand when they are heated, and this is also true of bulk nylon. On this basis as

the string is heated its natural length should increase (expansion); so it should not need to be

stretched so far to reach its stretched length, and so again it should tend to go flat.

But it doesn’t. It turns out that as the string is heated it is actually trying to contract along its length

(though it may get thicker), thereby increasing the length by which it is stretched and hence

tending to increase the string tension making it go sharp.

These two effects, the thermal contraction in the natural string length and the reduction in its

stiffness, work against each other but in this case the thermal contraction dominates and the

string actually goes sharp as the temperature rises.

This behaviour for nylon strings may turn out to be very different for gut strings, but we don’t know

yet.

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No Adjustment – Linear Density vs. Temperature

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There is also a third factor at work which is a change in the string’s mass.

As the temperature rises the absolute humidity, a measure of the mass of water in each cubic

metre of air, also tends to rise.

An increase in temperature tends to dry the string out, while an increase in humidity causes it to

absorb water.

Again there are two effects working in opposite directions.

The graph shows a plot of the string’s linear density – it’s mass per unit length – plotted against

temperature.

This particular string, which was tested at a low humidity, tended to gain mass as the temperature

(and absolute humidity) rose – but only very slightly. The total variation is only about 0.5% over a

40 °C temperature range.

The same string may behave differently at higher humidity levels; we don’t know yet.

A thinner 1st Octave A (A6) string, tested at both high and low humidity levels, showed the

opposite trend with a decrease in mass as the temperature rose – the string tending overall to dry

out as it got hotter. The variation in mass was also much larger at around 2.5% over a similar

temperature range.

9

Constant Frequency – Length Adjustment vs. Temperature

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Now we know a bit more about how the string actually behaves, what can we do about it?

One approach is to keep the string tuned to a constant frequency and see what adjustments are

required as the temperature changes.

This graph shows the tuning length adjustments (made by the motorised string winder) that were

required to keep the string tuned as the temperature was increased and decreased for several

cycles over a range of about 35 °C.

Notice that the size of the adjustments required are really very small – fractions of a millimetre.

We can fit a straight line to this response but a curve (quadratic) is better.

This curve can then be used to specify a tuning compensator which monitors the temperature and

adjusts the string length accordingly. These length adjustments should then match the

adjustments needed to keep the string in tune; but without actually needing to measure the string

frequency – which means that these tuning corrections could be made while the instrument is

being played.

10

Compensated using Length Adjustment vs. Temperature

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This graph shows what happened when this approach was used.

Again the string has been heated and cooled for several cycles over a range of about 35 °C, and

this time the string length has been adjusted based on the temperature and the curve obtained

from the previous test.

The frequency deviation has been significantly reduced, but the graph appears to be looping with

the frequency deviation gradually getting worse overall.

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Compensated using Length Adjustment vs. Temperature

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This graph shows the same frequency deviation data but as a function of time instead of temperature. This clearly shows that there is a steady drift in the string frequency, with the string gradually going flat.

The precise cause of this behaviour is not known, but it is probably a result of creep-related non-elastic behaviour in the string.

Each time the string tightens up and then relaxes, as the temperature rises and falls, it doesn’t return to the same point it was before but remains permanently stretched a bit more – gradually going flat.

This was probably the reason why the first response, for the unadjusted string, showed a zigzag pattern slowly moving to the right.

This approach, controlling the string length as a direct function of the temperature, would likely still require an electronic implementation, probably built on top of an automated tuning system – so it would still be expensive.

The automated tuning system would then be used to fully retune the harp (before a performance and at each interval perhaps) with the compensator making pre-emptive temperature-based adjustments between tunings.

The level of tuning control provided may be good enough, provided there were not strong temperature reversals (i.e. if the temperature was mainly just increasing or decreasing between tunings).

The same test with the thinner 1st Octave A (A6) string did not show this frequency drift. It may be that that string was better stabilised than the 3rd octave A string, even though both strings had been under tension on the test rig for some months before these tests were run (5 months for the 3rd Octave A string; 4 months for the 1st Octave A).

Gut strings may also behave differently, with better ‘memory’ – again we don’t know yet.

12

String Held at Constant Tension

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Instead of trying to control the string length adjustments directly we can try controlling the string

tension.

In this test the string tension has been held constant while the string has once again been heated

and cooled for several cycles over a range of about 35 °C.

The frequency deviation has been reduced even further than before and, instead of going sharp,

the string is now going slightly flat as the temperature rises, at a rate of about -0.1 ¢/°C; so even a

10 °C change in temperature would only result in a deviation of about 1 ¢.

As the string heats up and contracts, the compensator has to unwind the tuning peg to maintain

the constant tension. This adds a small amount of mass to the string, and this is why the string

now goes slightly flat as the temperature rises.

Notice that this time there does not appear to be any significant drift in the frequency over the

successive heating cycles; the graph is looping round and round. The gap between the top and

bottom of the loop (hysteresis) may be due to a thermal lag, with the string behaviour only

responding after a delay to a change from heating to cooling or vice versa, or it may be due to

some more fundamental property of the string.

So a good way to keep an aeolian harp in tune would be just to hang a fixed weight from the end

of each string. Not, however, very practical for a pedal or lever harp!

13

Compensated using Tension vs. Temperature

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During the constant frequency test discussed earlier, as well as measuring the required length

adjustment changes, we can also measure the tension changes required to keep the string in

tune.

In this case the required adjustments can be modelled quite well as just a straight line relating the

required tension to the temperature.

This graph shows the result of using this linear tension function to make pre-emptive tuning

adjustments based on the temperature.

The frequency deviation is better still, staying within a range of about 3 ¢, and is generally level

across the temperature range.

And again there is no significant drift in the frequency over the successive heating cycles.

Overall then we can get much better results controlling the string tension, rather than trying to

control the string length directly.

Sadly, trying to measure the string tension accurately over a range of temperatures is not easy,

making an electronic solution even more expensive.

14

Compensated using Tension vs. Length Adjustment

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In this final case the compensator function is maintaining the string tension as a linear function of

it’s length – combining what was learnt from the constant frequency test for both the required

tension and length adjustments.

If anything the frequency deviation is smaller still, staying within a range of about 2.5 ¢; it’s

generally level across the temperature range, and there is no significant drift in the frequency over

the successive heating cycles.

This approach has the advantage that the compensator should work effectively regardless of the

cause of the change in the string tension; whether it’s due to a change in temperature, humidity, or

even some residual string creep.

Also this form of compensation function, with tension varying linearly with length, corresponds to a

simple spring – which means that it may be possible to provide a relatively low cost mechanical

solution.

15

Possible Spring Compensators

StringString

Spring on Fixed AxleSpring on Turned Axle

Crank & Worm Gear

Pulley

Tuning Peg

The required compensator characteristics would probably require a clock-type spiral spring. This

slide shows two possible configurations.

On the left the string is connected directly to the outer end of the spring. The string is tuned by

turning the axle attached to the inner end of the spring, possibly using a worm gear arrangement

which would also provide a locking mechanism.

On the right the string runs over a pulley to a normal tuning peg, with the centre of the pulley now

attached to the end of the spring. This has the advantage that the inner end of the spring can be

fixed and the complex worm gear is not required. It would also be easier to change the string with

this approach. A possible disadvantage is that the thicker strings may require quite a large pulley

to move freely enough for the tuning compensation to be effective; with the pulleys having to be

overlapped in some way in order to maintain the normal string spacing.

In both cases steps must be taken to ensure the spring does not unwind significantly when the

string inevitably breaks.

The compensator mechanism must also not vibrate or buzz when the string is plucked.

16

Summary for 3rd Octave A

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Tension vs. Temperature3.1 ¢ over 35 °C & level

Tension vs. Length Adj.2.7 ¢ over 36 °C & level

Constant Tension4.5 ¢ over 36 °C @ −0.1 ¢/ °C

No Adjustment75 ¢ over 40 °C @ +2 ¢/ °C

Length Adj. vs. Temp.14 ¢ over 35 °C & drifting

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Linear Density rises

slightly with temperature

This slide gives a quick summary and comparison of the behaviour of this Nylon 3rd Octave A (A4)

Concert Harp string with no adjustment and with the four different tuning stabilisation methods

discussed.

In each case the range of the observed frequency deviation is given together with the

corresponding temperature range.

Where there is no adjustment made to the string, this can be thought of as being equivalent to

having a spring compensator with infinite spring stiffness. Conversely, holding the string at

constant tension is equivalent to using a spring compensator with zero stiffness. If the

corresponding frequency deviation versus temperature responses (the two graphs on the left-

hand side) have opposite gradients, with one going up and the other going down (as is the case

for this 3rd Octave A string), then a mechanical compensator with a spring of the right stiffness

connected in series with the string should be able to stabilise the string tuning (bottom right

graph).

17

Comparison with 1st Octave A

Tension vs. Temperature6.8 ¢ over 35 °C & level

Tension vs. Length Adj.8.8 ¢ over 35 °C & level

Constant Tension10 ¢ over 38 °C @ +0.2 ¢/ °C

No Adjustment69 ¢ over 39 °C @ +1.7 ¢/ °C

Length Adj. vs. Temp.10.5 ¢ over 37 °C & level

Linear Density falls

as temperature rises

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This slide gives a comparison with the results of similar tests run for a thinner Nylon 1st Octave A

(A6) Concert Harp string, again from Bowbrand.

These tests with the thinner string were carried out at a higher humidity level than those for the 3rd

Octave A string.

The unadjusted string again goes sharp as the temperature rises, but its linear density falls

suggesting that it is drying out and losing mass.

In the case where the string length is adjusted as a function of the temperature, the steady

frequency drift observed for the 3rd Octave A string is no longer present. It may be that this thinner

string was more settled with less residual creep.

In the constant tension case the string still goes sharp as the temperature rises, indicating that the

string mass lost through drying out must be greater than the additional string mass unwound from

the winding shaft to maintain the tension as the string attempts to contract.

Controlling the string tension as a function of the temperature or the string length adjustment

again gives the best results, though these are not quite as impressive as for the 3rd Octave A

string.

In both the no adjustment and constant tension cases the string goes sharp. Consequently a

mechanical spring compensator for this thinner string would require a spring with negative

stiffness, which is more complicated to implement. It would probably be more practical to

endeavour to maintain the string at close to constant tension using some form of spring and lever

arrangement such as that proposed by Lyles et al in patent number WO 2007/106600 A2

“Stringed Musical Instrument Using Spring Tension”.

18

Demonstrator

Constant ForceSpring: 79 N

Coil Spring to add required stiffness

Tension Adjustment for tuning

Bridge Pin

Pulley

A demonstrator has been built using a Gillian Weiss Cardboard Harp with nylon strings as the

platform.

A spring-based mechanical compensator has been implemented for the bottom string.

A single spring of the required specification was not available as a stock item, so instead a slightly

different approach has been taken using standard springs.

The bulk of the string tension is provided by a 79 Newton ‘constant force’ spring. This provides a

nearly constant tension.

Additional tension and spring stiffness is provided by adding a simple coil spring. Stretching the

coil spring enables the overall string tension, and hence its tuning, to be changed.

By changing, or even removing, the coil spring different spring characteristics can be tried.

Tests have shown that the best results for this particular string are obtained with the coil spring

removed. The next string up from the bottom has a similar length and gauge to the bottom string.

Typically, for a 10 °C increase in temperature, the frequency of this uncompensated string goes

up by about 10 ¢, while the frequency of the compensated string only increases by about 1 ¢.

Clearly the mechanical compensator approach has potential.

19

Caveats

What About Gut Strings?

Next Steps

Other Possibilities?

There had to be a but….

A simple mechanical spring compensator may not work for the thinner strings: tests with a 1st

octave A nylon string show that (due to the string drying out as it gets warmer) a mechanical compensator would require a spring with negative stiffness. On the other hand it is the lower non-metallic strings which most need tuning compensation. Adequate compensation could possibly still be applied to the thinner strings using some form of spring and lever arrangement holding the string at close to constant tension.

The tests with the 3rd Octave A string were carried out at low humidity and the behaviour may be different at higher humidity levels. Tests at low and high humidity with a thinner 1st Octave A nylon string, however, did not show dramatic differences in the string behaviour at the different humidity levels.

Any effects due to changes in the frame have so far been ignored.

The feel of the string may be different. This does not seem to be the case for the Cardboard Harp demonstrator, but the differences may be more noticeable on a tightly strung instrument.

Gut strings: Basically we don’t know until they are tested.

The mechanical spring compensator approach may still work very well.

An electronic solution controlling the string length directly may also be viable but again we don’t know yet.

Next steps: More tests with nylon strings; then gut strings and perhaps also ‘carbon’ strings.

The aim is to provide a proof of concept. Beyond this it would need the involvement of a harp maker to develop and test a prototype implementation.

Other ideas: Can the frequency of an unplayed string be measured more directly somehow?

Is there a relatively low-cost way to measure the string tension effectively?

Or maybe something completely different such as trying to surround the strings in a curtain of warm slightly moist air to provide a constant environment?

20

Acknowledgments

With thanks to:

Prof. Jim Woodhouse and Dr Claire Barlow of the Department of Engineering,

University of Cambridge, for their guidance and assistance during the course of this study.

Justin Grimwood – fabrication services and assistance: string winder and

bridge pin assemblies; cardboard harp demonstrator.

Dr Martin Oates – electronic circuits advice and suggestions.

Ian Papworth, Novatech Measurements Ltd – loadcell advice.

Jim Hinder, Kern-Liebers – constant force spring.

Jeanne Lynch-Aird for the use of her harps and advice on the tuning difficulties faced by harpists.

Dr Matteo Frigo and Prof. Steven G. Johnson for making their FFTW

algorithm freely available ( www.fftw.org ).