Stringed instruments are sensitive to temperature and humidity changes. In particular, plain gut or nylon strings on a harp can be sufficiently sensitive to environmental changes as to become noticeably out of tune during the course of a performance. For instruments like the violin, the player can make corrections by changing their fingering. For open-string instruments such as the harp, though, the player can only wait for a break in the music to retune an errant string.
Automated string tuning systems have been developed [1
], but these generally require the instrument to be switched into a specific tuning mode, with the string being plucked or struck in order to generate a tone that can be compared with the required musical pitch. A motor is then used to adjust the string tension as necessary. Even if such a system were adapted for use during a performance, it would not help with the situation where the string is already noticeably out of tune when it next comes to be played. Furthermore, any such system will always require multiple tuning iterations for each string because the adjustment scaling factor, used to convert the measured tuning error into the required string winding adjustment, will never be known exactly.
A study of the mechanical properties of centreless-ground (“rectified”) nylon harp strings has shown that holding such strings at a constant tension may be enough to keep them acceptably in tune [3
]. Many players, however, prefer natural gut strings, which are much more sensitive to changes in humidity and can change in unpredictable ways: on the same instrument, a given gut string may go sharp while an adjacent string goes flat. Ideally, a tuning control scheme needs some way of measuring the pitch of a string without the need for that string to be plucked and without making an audible noise that would disrupt a performance going on around the instrument. Tuning corrections could then be made repeatedly, and discreetly, maintaining the string at the correct pitch and ready for use.
The simplest way to monitor an individual string, without interfering with its playing properties, is to make the necessary measurements at one or other of the string’s terminations. Some kind of actuator would be used to apply a low-level mechanical input, and some kind of sensor would measure the response. Ideally, these functions would be combined in the same device, which would also be inexpensive and robust. Such a system is described and tested in this paper, based on a low-cost piezoelectric disc transducer. Electric and electro-acoustic harps often use piezoelectric pickups, with each string passing over a separate pickup at the point where the string meets the soundboard. However, these pickups have a different structure from the transducer used here, and it will be demonstrated in Section 4.7
that they are less effective for the purposes of monitoring tuning in the manner explored here.
As well as exploring the ability to measure the string pitch, the study also investigates the effects of varying the string stress and the use of different drive levels. An approximate theoretical model is developed and compared with measurements; and the performance of a possible tuning system is demonstrated. Finally, some consideration is also given to other possible pitch measurement approaches based on the use of piezoelectric transducers.
2. Methods and Materials
2.1. Test Rig
A simple test rig (Figure 1
) was constructed on which a string could be mounted approximately 13 mm above a wooden baseboard. A motorised string winder was mounted at one end of the string, allowing precise string length adjustments to be made when required. From the winding shaft, the string passed a fixed vertical bridge pin 5.5 mm in diameter. At its other end the string passed over a small bridge resting on the piezoelectric transducer, and then sloped down to a clamping point on the baseboard. The bridge consisted of a semi-cylindrical steel spacer, 4 mm in length, holding the string about 2.5 mm above the transducer disc. The mass of the spacer was about 0.2 g. The termination points at each end of the vibrating length of string (the vertical bridge pin and the horizontal spacer) were set at 90° to each other so that the effective vibrating length of the string was not affected by the polarisation direction of the transverse vibration [3
The string used for these tests was a centreless-ground nylon string from the Bowbrand “Pedal Nylon” range, with an unstretched diameter of 0.84 mm and an unstretched linear density of 0.593 g/m. The string vibrating length L between the vertical bridge pin and the piezoelectric transducer was 223 mm. The length of the string tail section, from the piezoelectric transducer down to the end clamp attached to the baseboard, was 66 mm, with a break angle of about 3°.
For the main tests to be described, the transducer used was a low-cost disc unit marketed as a piezoelectric sounder (Maplin QY13P [5
]). This consists of a circular metal base plate 27 mm in diameter and 0.3 mm thick, with a piezoelectric disc 20 mm in diameter and 0.2 mm thick bonded to one face. The base plate forms one electrode, while a thin metal coating on the other side of the piezoelectric disc provides the second electrode. This type of transducer is designed so that the piezoelectric layer contracts or expands radially in response to an applied electrical voltage, causing the base plate to bend in an axisymmetric manner: it ‘domes’ in and out, similar to the fundamental mode of vibration of a circular diaphragm. To allow this motion to occur unimpeded, the transducer was supported on an annular cup washer with a cavity under most of the base plate. Direct measurements with a laser-Doppler vibrometer confirmed that the transducer did indeed move in the expected direction, normal to the plane of the base plate [6
]. Figure 2
shows the piezoelectric transducer mounted on its annular washer, with the string running over the semi-cylindrical spacer positioned in the centre of the top electrode.
Following an initial series of investigations, the test rig was left for a period of about six months prior to taking the set of measurements reported here. This had the advantage of allowing plenty of time for the string to settle. While variations in the string tension and frequency could be expected to result from changes in the ambient temperature [3
], the string was (at least for the purpose of these tests) no longer creeping. To take advantage of this stability, the test sequence was organised to complete as many measurements as possible before making any tuning adjustments.
2.2. Electrical Measurements
In operation, a sinusoidal voltage was applied to the transducer, producing normal motion which drove the string into vibration. That vibration resulted in a reaction force at the bridge, causing additional axisymmetric bending motion in the base plate of the transducer (which acts as a spring). This in turn produced strain in the piezoelectric layer, which has the effect of modulating the capacitance of the piezoelectric transducer. This modified capacitance can be monitored via an electrical measurement.
The electrical impedance
of the piezoelectric transducer was measured as a function of frequency using the bridge circuit shown in Figure 3
. In order to select the reference capacitance
, some idea of the transducer capacitance was required. Initial measurements were made using a home-built capacitance meter which applied positive and negative voltage steps, via a reference resistor, and measured the rise and fall times of the voltage across the capacitance. This had a test cycle frequency of about 42 Hz, so the values obtained should be close to the DC capacitance for the piezoelectric transducer. Using this approach gave a capacitance of around 23–24 nF when the transducer was removed from the rig and able to move freely. The reference impedance for the other arm of the bridge was therefore selected to be a capacitor of about the same size, with a nominal value of 22 nF, and a measured value (using the same capacitance meter) of 26 nF.
For the intended test frequency range the impedance of both the test and reference capacitors was expected to be in the region of 10 k. The resistive elements of the bridge, and , were chosen accordingly. To allow easy exploration of the effects of variations in the bridge balance, these two components were provided using a single 20 k linear potentiometer. The instrumentation amplifier used for measuring the voltage difference across the bridge (Texas Instruments INA121P) was chosen for its very high input impedance () and the ability to set its gain using a single external resistor. A bias current path to ground was required for each input, and two 10 M resistors, and , were included for that purpose. A second instrumentation amplifier was connected directly across the piezoelectric transducer for use in recording manual pluck responses from the string. The plucks were made using a plastic guitar plectrum.
A two-channel digital oscilloscope and function generator (Velleman PCSGU250) was used both to provide the input signal to the impedance bridge, and to measure and record the output. While having limited input resolution and sampling duration, this low-cost unit had the advantage of being straightforward to operate under software control. One oscilloscope channel was used to measure the bridge output, while the other channel was switched between recording the output of the signal generator, when measuring the transducer impedance, and the output of the second instrumentation amplifier, when recording pluck responses.
As will be explained in Section 4.3
, it is beneficial to measure the bridge output relative to the input signal in complex form. Sine functions were fitted to the recorded waveforms for the bridge input and output using a least-squared-error approach: a function
of known frequency
can be fitted as
. The magnitudes and phases of the fitted functions were then combined to give the complex output.
Tests were also run using a simple voltage divider configuration, with just the reference and test impedances connected in series. In this configuration, however, the overall testing sensitivity is seriously limited by any gain differences in the instrumentation amplifiers, oscilloscope input circuits or analogue-to-digital converters used to measure the voltages across the two impedances. The bridge circuit, while being somewhat more complex to analyse, was found to give significantly better performance.
5. Discussion and Conclusions
This study has shown that the transverse resonance frequencies of a stretched flexible string can be determined from the electrical impedance response of a piezoelectric transducer in contact with the string at one of its endpoints. The successful measurements were made using a low-cost circular piezoelectric transducer, 27 mm in diameter, designed to flex its base plate. The transducer is driven with a sinusoidal waveform at a low level, inducing some barely-audible vibration in the string. The vibrating string then exerts force back on the transducer, resulting in a modulation of its capacitance which can be detected using a bridge circuit. By sweeping the sinusoid over a range near the fundamental resonance of the string, a peak at the resonance frequency can be detected.
The capacitance of a piezoelectric transducer, below its own mechanical resonance, can be expected to vary noticeably with temperature due to the thermal sensitivity of the dielectric constant of the piezoelectric material [7
]. Consequently, the impedance measurement bridge will usually be operating somewhat out of balance. The result is to obscure the precise frequency of the string resonance. However, through the development of a simple explanatory model for the transducer behaviour, a straightforward approach has been demonstrated for correcting the bridge balance in software. The string resonance can then be robustly and accurately identified.
It has also been demonstrated that this frequency measurement approach can be used as the basis for a tuning control mechanism, capable of maintaining a musical instrument string at the required pitch, without the string having to be plucked or played. An instrument equipped with such a system would be stabilised against changes in both the strings and in the instrument frame and soundboard. This would be of direct benefit to musicians, and to harp players in particular, who have no other means to adjust a mistuned string during a performance.
Tests using a much smaller Camac harp pickup, designed to work with small bulk strains rather than by flexure, were not able to distinguish the string resonance via the transducer impedance response. This suggests that a quite different transducer design will be required from that normally used for the pickups in electric and electro-acoustic harps. This need not be a drawback, since it seems likely that the transducers for auto-tuning purposes would work best if mounted on a base that is as solid and inert as possible: this would reduce any influence of vibration of the base structure and also minimise cross-coupling between adjacent strings. This suggests positioning the transducers at the far end of the string, away from the soundboard.
A brief attempt was also made to measure the string resonance by injecting tones at one end of the string and measuring the response amplitude at the other end. While the resonance could be identified, as a peak in the output level, the drive levels required for successful operation were much higher than with the impedance measurement approach. Such an approach would have the additional drawback of requiring a separately wired transducer at each end of the string. Most electric or electro-acoustic harps that have a separate piezoelectric pickup on each string still have multiple pickups connected in parallel to a common amplifier circuit.
On the pedal or lever harp, there would be a potential issue with the fourchettes or other semitone mechanisms, mounted along the neck of the instrument. These work by gripping the string, slightly shortening its vibrating length and hence raising the sounding pitch. A number of possible solutions may be available depending on how the tuning control system is to be used. In concert performances, there are often long periods when the harp is not being played. One possibility, therefore, would be for the player to disengage the semitone controls (pedals or levers) so that the tuning control mechanism can operate on the open strings until the harp is to be played.
If it is desired that the tuning control mechanism should be in more continuous use, then a means will be needed to deactivate it when the string is played. This could perhaps be achieved by detecting large-amplitude string excitation, or by using sensors to detect when the player’s hands or fingers are near the strings. A delay would have to be allowed before reactivating the tuning control to allow the string vibration to decay. The existing mechanical pedal and lever mechanisms could then be replaced by electro-mechanical servo units, which automatically disengage the fourchettes, or equivalent, when the tuning control is active. Alternatively, the piezoelectric transducers could be set to make contact with the string below the fourchettes, and engaged or disengaged under servo control. The latter option might allow the tuning control system to be retrofitted to an existing instrument. Although these suggestions all sound rather complicated, it should be borne in mind that the cost of a full-size concert harp can run into tens of thousands of dollars. While considerable development effort would be required, all the necessary components are readily available at relatively low cost.
During the course of a performance, the reference pitch of an orchestra can change, but the harp player has no means to follow such changes. A system capable of making automated tuning adjustments, however, could also have its reference pitch adjusted (perhaps manually by the player), causing the whole instrument to be retuned accordingly.
A further problem that may be encountered by any tuning control system operating only on the string fundamental stems from the fact that, in practice, there are always two “fundamental frequencies”. Real strings and their associated termination geometry are never perfectly symmetric, so that the two polarisations of string vibration will always have slightly different resonance frequencies [4
]. On the test rig, efforts were made to minimise the influence of these split peaks; but an automated tuning control system on a real instrument would need to cope with this phenomenon.