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An electronic aid to the tuning of percussive musical instruments
J. Sound
Vz%. (1967)
6 (2), 180-186
AN ELECTRONIC
AID TO
MUSICAL
THE
TUNING
OF PERCUSSIVE
INSTRUMENTS-/E. V. VERNON
Department of Electronics, The University, Southampton, England (Received 27 October 1966) An apparatus is described which will determine the frequency of an audio signal by measuring the time of a sample of sixteen cycles. This makes the apparatus particularly suitable for determining the frequency of heavily damped notes of musical instruments, such as pianos. An accuracy of about +o.I% in the bass and + 0.2% in the treble, that is about one-hundredth of a tone, has been achieved.
I. INTRODUCTION A well-trained musical ear is capable of detecting when the frequency of a particular note of a musical instrument is mistuned by as little as about four-hundredths of a semitone (four cents). This means that, for instance, middle C (261 Hz) on a piano needs to be tuned to within about + 0.6 Hz to be considered to be “ in tune “. An even more demanding requirement is that the unison of a piano tricord at this frequency should be accurate within about & 0.1 Hz to reduce beat frequency effects to an acceptable level.1 To achieve this order of accuracy a conventional frequency counter would need to count over a period of IO set, during the whole of which time the signal/noise level would have to be much greater than unity. Few percussive musical instruments, used under normal conditions, fulfil this requirement; in almost all cases the signal/noise level is below one well before the IO set have passed. An apparatus has been designed and built which will give the required accuracy, and better, by accurately timing the length of a pulse from a short sample of the signal obtained from the musical instrument. In outline, the musical note is picked up by a sensitive microphone, and the resulting electric signal is amplified and then fed through a frequency selective amplifier so that only the overtone it is required to measure is allowed through. The signal is then fed to a Schmitt trigger circuit to produce a square wave. This is divided down by a number of “scale-of-two ” divider circuits, the number depending on the accuracy of the result required, and then fed to a circuit which will, on triggering, allow one pulse only to pass through it. The duration of this pulse is then measured on a conventional electronic timer. 2.
APPARATUS
Figure I is a block diagram which shows the principle of the method. As, in essence, the apparatus measures the time between two successive crossings of the zero axis of the signal at A, it is necessary that the signal is free from even harmonics generated in the amplifier and all but the wanted overtone from the incoming signal. The latter is because it is known that there is inharmonicity among the overtones of a vibrating piano string [ I, z], and also to a greater or lesser degree in other musical instruments. It is also necessary that t Provisional patent no. 14739.
1 Since this paper was written, Dr. W. H. George has drawn the author’s attention to a paper on the tuning of unisons, by Kirk [4]. 180
181
ELECTRONIC TUNER FOR MUSICAL INSTRUMENTS
the signal/noise ratio is very high as small noise voltages will slightly alter the crossing points on the time axis. A sensitive microphone, an S.T.C. Type 4038A (B.B.C. pattern), was chosen, not only because of its sensitivity but also because of its good low frequency.response. The output
II Signal level meter
Microphone
LOW noise amplifier
Tuneable frequency selectwe amplifier
-
A
I Schmitt trigger
“Square pulse to two positive pulses” converter
Swtch
Figure I.
I
I
.
Four bistoble stages
-
Commercial psec timer
’
Block diagram showing principle of method.
from the microphone is fed to a single stage low noise amplifier (Figure z), which uses a This amplifier is followed by a conventional BC 109 NPN planar silicon transistor. emitter follower stage so that the frequency selective amplifier which follows is fed from a low impedance source. The selective amplifier, which can be tuned over the range 20 Hz to IO kHz, has a Q of at least 60 ; this Q is sufficient to eliminate unwanted overtones except that when measurements are made on small pianos the amplitudes of the fundamental frequencies of notes below about 50 Hz are too low with respect to the amplitudes
output I
1 -L
Figure z. Low noise amplifier.of the overtones. Considerable second harmonic is then passed, causing errors in the final readings. Details of the selective amplifier are to be published separately [3]. As the maximum undistorted output from the selective amplifier is of the order of 0.5 v only, the signal is further amplified so that the small voltage, about 0.1 v, required to operate the Schmitt trigger represents only a very small fraction of the signal amplitude. This reduces errors due to harmonic distortion. The circuits of the amplifier, followed by an emitter follower, and the Schmitt trigger are shown in Figure 3. A meter and associated circuit is connected to the emitter follower to indicate the signal amplitude, as it is necessary for the amplitude to be above a certain level for the Schmitt trigger circuit to operate
182
E. V. VERNON
satisfactorily. The emitter follower is directly coupled to the Schmitt trigger and the high stability 2.2 kQ and 68 kL? resistors are chosen to give a one-to-one mark/space ratio in the trigger output waveform. The Schmitt trigger is followed by four cascaded bistable “ divide by two ” circuits, so that a sample of sixteen cycles of the incoming signal to the
Figure
3.
Amplifier, emitter follower and Schmitt trigger circuit.
microphone produces one cycle of a square wave at the output. The circuit of a typical bistable is given in Figure 4. The next block of circuitry, Figure 5, is designed to allow a particular square pulse from the square wave through it, the particular pulse being the one arriving at the circuit immediately after closing the switch connected to the Schmitt trigger of this circuit. The duration of the pulse is eight times that of the output signal from the selective amplifier.
*output
I0k.Q
*I2
OA81 470 pF.. II
Input
=
OAI31 6.8 k.2
3680 ::P
0100 -jcF
::6 8 :; kJ2
u I.
1 &
6
Figure
4.
The circuit of a b&able
470 pF
stage.
The method of operation of the circuit is as follows. When the switch S1 is closed the Schmitt circuit triggers and the transistor T2 (see Figure 6) switches hard on. Therefore, a positive going pulse is produced at its output (A). The diode across the output prevents a negative going pulse being produced at the output when the switch is opened. The positive pulse is then fed to bistable 1 at II (Figure 4). (N.B. The bistables used for the circuits in Figure 5 are similar to those of Figure 4, except that there are two inputs, at II
ELECTRONIC
TUNER FOR MUSICAL
INSTRUMENTS
‘83
and I,; the other input and the two 470 pF capacitors are omitted.) The positive pulse switches bistable 1 so that the collector of the transistor T1 (Figure 4) is at about - 15 v. Square wove train imput
D
PiNO
w gate
.
Emitter follower ond differentntor
B
1
Figure
Single square pulse output
AND
gate 2
5. Block diagram
of circuit
.
: t Emitter follower and differentiator
E
c
Bistable 2
-
to select one square pulse from a train of square waves. * -15v
330 pF
Figure
6.
Schmitt
trigger
circuit
for single pulse selector.
r--+‘5v
lnput2 _onslDz Figure
7. Circuit
/--output of AND gate.
Now, the AND gates in Figure 5 will pass square waves fed to input 1 (Figure 7) if the voltage on the diode II2 is considerably more negative than - 6 v. AND gate 1 (Figure s), will therefore open when diode D2 is biased to about - 15 v by bistable 1, and a square pulse will appear at B. This is fed to an emitter follower and differentiator (Figure 8),
184
E. V. VERNON
which converts the square pulse to a positive going pulse. (The negative going pulse is removed by the diode across the output.) This positive going pulse performs two functions ; it switches bistable 1, consequently closing AND gate 1 so that only one square pulse is allowed through the gate, and switches bistable 2, opening AND gate 2. When AND gate 2 opens, a square pulse appears in the output (Figure s), and also a positiye going pulse is produced at E. This pulse switches bistable 2 and hence AND gate 2 closes. This means ,r-15v
-L oc42
Input
2000
2000
pF
pF
T-T---““+p”+
-----YFigure
8. Circuit
of emitter
follower
and differentiator.
Squore wove imput from bistables Time_ Schmitt trigger output A (switch closed ot XI
II
X
Output from bistoble 1
Woveform
Waveform
ot B
ot C
;l 11
Bistoble 2 output at D
output from AND gote 2 Waveform
Figure
g. Voltage
ot f
i-1
11
waveforms
at various
points in the circuit
of Figure
5.
that only one square pulse appears in the output each time the Schmitt trigger switch is closed. The waveforms at various parts of the circuit of Figure 5 are shown in Figure 9. If automatic readings on the timer are required, the whole of the circuitry of Figure 5 can be bypassed by switching Sz to X (Figure IO). The remaining circuitry converts the single square pulse output of Figure 5 into two positive going pulses of a few volts amplitude, sufficient to operate a conventional electronic timer (in this case a Racal type SA535 set to count in psec). The output from AND gate 2 is then fed to a clipper circuit, shown at the left-hand side of Figure IO, which removes
ELECTRONIC
TUNER FOR MUSICAL
INSTRUMENTS
185
any traces of the unwanted square wave train which get through the gate because of the finite reverse resistance of the diodes in this gating circuit. The square pulse is then fed to a differentiator network and on to a phase splitter. The square pulse therefore gives rise to two positive and two negative going pulses of equal amplitude at the emitter and collector of the phase splitter. The diodes connected to the emitter and collector are so biased that only the negative going pulses are passed. These two pulses are amplified, becoming positive going pulses, and are fed to the electronic timer. The first pulse switches the timer on, and the second switches it off.
I-‘5v I0k.Q
IkJL
33k.Q
To imput of single pulse selector -= circuit
OA91
0891
Figure
IO.
470 k.Q
5.6 kJ2
0.1 fiF
il-
0
II
4-
>-To timer
0.1 t”F
Clipper circuit and square pulse to positive pulse converter.
The timer then indicates the time between the two positive of half a period of the square wave output from the bistables. original incoming signal is then given by
f=
(timer
8 x 10~ HZ. reading in psec)
pulses, that is, the length The frequency, f, of the
(I)
3. ACCURACY
The overall accuracy of the instrument was determined by the following method. A speaker was excited by the amplified output from a calibrated crystal oscillator, and the audio signal was fed to the microphone. The timer readings were found to be correct to within & 2 psec for the oscillator frequencies used, over the frequency range IOO Hz to IO kHz, provided that the audio signal was above a certain level. The reading on the meter, connected to A (Figure I), corresponding to this level was noted, and all frequency measurements were made at signal levels above this minimum. Equation (I) can be modified to take into account the timing error, as follows : f=
(timer
8x reading
10~
-
in psec) f
2
Hz.
Hz the error is k 0.003 Hz, and at IO kHz it is * 0.3 Hz. if the incoming signal is a pure sine wave, or if the overtones are exact integral multiples of the fundamental, but it has already been mentioned that the upper partials of many instruments are inharmonic, and so the accuracy now depends on the efficiency with which the unwanted overtones can be suppressed. Therefore This
13
at a frequency
accuracy
of
IOO
is only obtained
186
E. V. VERNON
The unwanted overtones were suppressed by the variable frequency selective amplifier, details of which will be published elsewhere. This amplifier could be tuned over the range 20 Hz to IO kHz and the Q was about 60. An improved version of this amplifier gives values of Q up to 200, but it has not yet been used with the electronic tuner. The accuracy of the results, as indicated by the spread in the readings of the time using a piano as the source, has been found to depend on the piano note struck. With a small upright piano the error results obtained from some twenty readings of each of three notes are given in Table I. TABLE I Note
c,,,
c
C”
Frequency (Hz) Error : (a) as a percentage (b) in cents
32’7
261
2993
+0*14 &2
+ 0.03 + 25
+-0’02 k2.5
The error in the measurement of the lowest note is undoubtedly due to the presence of some of the first overtone in the signal after feeding it through the selective amplifier. (The length of the string for C,,, was about I m ; the amplitude of the fundamental was very much less than that of the first overtone.) 4. CONCLUSION
The instrument has been shown to have the required accuracy for tuning purposes, but a small piano tuned exactly to the equal temperament scale was found to appear to be sharp at the bass end and flat at the treble end. This effect is undoubtedly due to the inharmonicity of the overtones, and so work on the measurement of the inharmonicity of some strings in a number of pianos is now being carried out. It is believed by the author that this will throw some light on the amount of deviation from the equal temperament scale necessary to achieve a musically satisfactory tuning. ACKNOWLEDGMENTS The author is indebted to Professor G. D. Sims, Head of the Electronics Department, Southampton University, for his encouragement, and for the facilities provided. He is also indebted to many colleagues for helpful discussion, and, in particular, to Mr. K. G. Nichols for the design of the logic circuits of the single pulse selector. The author acknowledges the co-operation of Professor P. A. Evans, Head of the University Music Department, and his colleagues. Finally the author wishes to thank Mr. B. G. Feasey, a former student of the Electronics Department, who did much of the original design as his final year project. REFERENCES I. R. W. YOUNG1954 Acustica, 4, 259. Inharmonicity of piano strings. E. DONNELL BLACKHAM 1965 Scient. Am. 213,88. The physics of the piano. 3. W. R. KNOWLES,S. W. PUNNETT and E. V. VERNON (to be published) An improved tunable audio frequency amplifier. 4. R. E. KIRK 19593. acoust. sot. Am. 31, 1644. 2.
Vz%. (1967)
6 (2), 180-186
AN ELECTRONIC
AID TO
MUSICAL
THE
TUNING
OF PERCUSSIVE
INSTRUMENTS-/E. V. VERNON
Department of Electronics, The University, Southampton, England (Received 27 October 1966) An apparatus is described which will determine the frequency of an audio signal by measuring the time of a sample of sixteen cycles. This makes the apparatus particularly suitable for determining the frequency of heavily damped notes of musical instruments, such as pianos. An accuracy of about +o.I% in the bass and + 0.2% in the treble, that is about one-hundredth of a tone, has been achieved.
I. INTRODUCTION A well-trained musical ear is capable of detecting when the frequency of a particular note of a musical instrument is mistuned by as little as about four-hundredths of a semitone (four cents). This means that, for instance, middle C (261 Hz) on a piano needs to be tuned to within about + 0.6 Hz to be considered to be “ in tune “. An even more demanding requirement is that the unison of a piano tricord at this frequency should be accurate within about & 0.1 Hz to reduce beat frequency effects to an acceptable level.1 To achieve this order of accuracy a conventional frequency counter would need to count over a period of IO set, during the whole of which time the signal/noise level would have to be much greater than unity. Few percussive musical instruments, used under normal conditions, fulfil this requirement; in almost all cases the signal/noise level is below one well before the IO set have passed. An apparatus has been designed and built which will give the required accuracy, and better, by accurately timing the length of a pulse from a short sample of the signal obtained from the musical instrument. In outline, the musical note is picked up by a sensitive microphone, and the resulting electric signal is amplified and then fed through a frequency selective amplifier so that only the overtone it is required to measure is allowed through. The signal is then fed to a Schmitt trigger circuit to produce a square wave. This is divided down by a number of “scale-of-two ” divider circuits, the number depending on the accuracy of the result required, and then fed to a circuit which will, on triggering, allow one pulse only to pass through it. The duration of this pulse is then measured on a conventional electronic timer. 2.
APPARATUS
Figure I is a block diagram which shows the principle of the method. As, in essence, the apparatus measures the time between two successive crossings of the zero axis of the signal at A, it is necessary that the signal is free from even harmonics generated in the amplifier and all but the wanted overtone from the incoming signal. The latter is because it is known that there is inharmonicity among the overtones of a vibrating piano string [ I, z], and also to a greater or lesser degree in other musical instruments. It is also necessary that t Provisional patent no. 14739.
1 Since this paper was written, Dr. W. H. George has drawn the author’s attention to a paper on the tuning of unisons, by Kirk [4]. 180
181
ELECTRONIC TUNER FOR MUSICAL INSTRUMENTS
the signal/noise ratio is very high as small noise voltages will slightly alter the crossing points on the time axis. A sensitive microphone, an S.T.C. Type 4038A (B.B.C. pattern), was chosen, not only because of its sensitivity but also because of its good low frequency.response. The output
II Signal level meter
Microphone
LOW noise amplifier
Tuneable frequency selectwe amplifier
-
A
I Schmitt trigger
“Square pulse to two positive pulses” converter
Swtch
Figure I.
I
I
.
Four bistoble stages
-
Commercial psec timer
’
Block diagram showing principle of method.
from the microphone is fed to a single stage low noise amplifier (Figure z), which uses a This amplifier is followed by a conventional BC 109 NPN planar silicon transistor. emitter follower stage so that the frequency selective amplifier which follows is fed from a low impedance source. The selective amplifier, which can be tuned over the range 20 Hz to IO kHz, has a Q of at least 60 ; this Q is sufficient to eliminate unwanted overtones except that when measurements are made on small pianos the amplitudes of the fundamental frequencies of notes below about 50 Hz are too low with respect to the amplitudes
output I
1 -L
Figure z. Low noise amplifier.of the overtones. Considerable second harmonic is then passed, causing errors in the final readings. Details of the selective amplifier are to be published separately [3]. As the maximum undistorted output from the selective amplifier is of the order of 0.5 v only, the signal is further amplified so that the small voltage, about 0.1 v, required to operate the Schmitt trigger represents only a very small fraction of the signal amplitude. This reduces errors due to harmonic distortion. The circuits of the amplifier, followed by an emitter follower, and the Schmitt trigger are shown in Figure 3. A meter and associated circuit is connected to the emitter follower to indicate the signal amplitude, as it is necessary for the amplitude to be above a certain level for the Schmitt trigger circuit to operate
182
E. V. VERNON
satisfactorily. The emitter follower is directly coupled to the Schmitt trigger and the high stability 2.2 kQ and 68 kL? resistors are chosen to give a one-to-one mark/space ratio in the trigger output waveform. The Schmitt trigger is followed by four cascaded bistable “ divide by two ” circuits, so that a sample of sixteen cycles of the incoming signal to the
Figure
3.
Amplifier, emitter follower and Schmitt trigger circuit.
microphone produces one cycle of a square wave at the output. The circuit of a typical bistable is given in Figure 4. The next block of circuitry, Figure 5, is designed to allow a particular square pulse from the square wave through it, the particular pulse being the one arriving at the circuit immediately after closing the switch connected to the Schmitt trigger of this circuit. The duration of the pulse is eight times that of the output signal from the selective amplifier.
*output
I0k.Q
*I2
OA81 470 pF.. II
Input
=
OAI31 6.8 k.2
3680 ::P
0100 -jcF
::6 8 :; kJ2
u I.
1 &
6
Figure
4.
The circuit of a b&able
470 pF
stage.
The method of operation of the circuit is as follows. When the switch S1 is closed the Schmitt circuit triggers and the transistor T2 (see Figure 6) switches hard on. Therefore, a positive going pulse is produced at its output (A). The diode across the output prevents a negative going pulse being produced at the output when the switch is opened. The positive pulse is then fed to bistable 1 at II (Figure 4). (N.B. The bistables used for the circuits in Figure 5 are similar to those of Figure 4, except that there are two inputs, at II
ELECTRONIC
TUNER FOR MUSICAL
INSTRUMENTS
‘83
and I,; the other input and the two 470 pF capacitors are omitted.) The positive pulse switches bistable 1 so that the collector of the transistor T1 (Figure 4) is at about - 15 v. Square wove train imput
D
PiNO
w gate
.
Emitter follower ond differentntor
B
1
Figure
Single square pulse output
AND
gate 2
5. Block diagram
of circuit
.
: t Emitter follower and differentiator
E
c
Bistable 2
-
to select one square pulse from a train of square waves. * -15v
330 pF
Figure
6.
Schmitt
trigger
circuit
for single pulse selector.
r--+‘5v
lnput2 _onslDz Figure
7. Circuit
/--output of AND gate.
Now, the AND gates in Figure 5 will pass square waves fed to input 1 (Figure 7) if the voltage on the diode II2 is considerably more negative than - 6 v. AND gate 1 (Figure s), will therefore open when diode D2 is biased to about - 15 v by bistable 1, and a square pulse will appear at B. This is fed to an emitter follower and differentiator (Figure 8),
184
E. V. VERNON
which converts the square pulse to a positive going pulse. (The negative going pulse is removed by the diode across the output.) This positive going pulse performs two functions ; it switches bistable 1, consequently closing AND gate 1 so that only one square pulse is allowed through the gate, and switches bistable 2, opening AND gate 2. When AND gate 2 opens, a square pulse appears in the output (Figure s), and also a positiye going pulse is produced at E. This pulse switches bistable 2 and hence AND gate 2 closes. This means ,r-15v
-L oc42
Input
2000
2000
pF
pF
T-T---““+p”+
-----YFigure
8. Circuit
of emitter
follower
and differentiator.
Squore wove imput from bistables Time_ Schmitt trigger output A (switch closed ot XI
II
X
Output from bistoble 1
Woveform
Waveform
ot B
ot C
;l 11
Bistoble 2 output at D
output from AND gote 2 Waveform
Figure
g. Voltage
ot f
i-1
11
waveforms
at various
points in the circuit
of Figure
5.
that only one square pulse appears in the output each time the Schmitt trigger switch is closed. The waveforms at various parts of the circuit of Figure 5 are shown in Figure 9. If automatic readings on the timer are required, the whole of the circuitry of Figure 5 can be bypassed by switching Sz to X (Figure IO). The remaining circuitry converts the single square pulse output of Figure 5 into two positive going pulses of a few volts amplitude, sufficient to operate a conventional electronic timer (in this case a Racal type SA535 set to count in psec). The output from AND gate 2 is then fed to a clipper circuit, shown at the left-hand side of Figure IO, which removes
ELECTRONIC
TUNER FOR MUSICAL
INSTRUMENTS
185
any traces of the unwanted square wave train which get through the gate because of the finite reverse resistance of the diodes in this gating circuit. The square pulse is then fed to a differentiator network and on to a phase splitter. The square pulse therefore gives rise to two positive and two negative going pulses of equal amplitude at the emitter and collector of the phase splitter. The diodes connected to the emitter and collector are so biased that only the negative going pulses are passed. These two pulses are amplified, becoming positive going pulses, and are fed to the electronic timer. The first pulse switches the timer on, and the second switches it off.
I-‘5v I0k.Q
IkJL
33k.Q
To imput of single pulse selector -= circuit
OA91
0891
Figure
IO.
470 k.Q
5.6 kJ2
0.1 fiF
il-
0
II
4-
>-To timer
0.1 t”F
Clipper circuit and square pulse to positive pulse converter.
The timer then indicates the time between the two positive of half a period of the square wave output from the bistables. original incoming signal is then given by
f=
(timer
8 x 10~ HZ. reading in psec)
pulses, that is, the length The frequency, f, of the
(I)
3. ACCURACY
The overall accuracy of the instrument was determined by the following method. A speaker was excited by the amplified output from a calibrated crystal oscillator, and the audio signal was fed to the microphone. The timer readings were found to be correct to within & 2 psec for the oscillator frequencies used, over the frequency range IOO Hz to IO kHz, provided that the audio signal was above a certain level. The reading on the meter, connected to A (Figure I), corresponding to this level was noted, and all frequency measurements were made at signal levels above this minimum. Equation (I) can be modified to take into account the timing error, as follows : f=
(timer
8x reading
10~
-
in psec) f
2
Hz.
Hz the error is k 0.003 Hz, and at IO kHz it is * 0.3 Hz. if the incoming signal is a pure sine wave, or if the overtones are exact integral multiples of the fundamental, but it has already been mentioned that the upper partials of many instruments are inharmonic, and so the accuracy now depends on the efficiency with which the unwanted overtones can be suppressed. Therefore This
13
at a frequency
accuracy
of
IOO
is only obtained
186
E. V. VERNON
The unwanted overtones were suppressed by the variable frequency selective amplifier, details of which will be published elsewhere. This amplifier could be tuned over the range 20 Hz to IO kHz and the Q was about 60. An improved version of this amplifier gives values of Q up to 200, but it has not yet been used with the electronic tuner. The accuracy of the results, as indicated by the spread in the readings of the time using a piano as the source, has been found to depend on the piano note struck. With a small upright piano the error results obtained from some twenty readings of each of three notes are given in Table I. TABLE I Note
c,,,
c
C”
Frequency (Hz) Error : (a) as a percentage (b) in cents
32’7
261
2993
+0*14 &2
+ 0.03 + 25
+-0’02 k2.5
The error in the measurement of the lowest note is undoubtedly due to the presence of some of the first overtone in the signal after feeding it through the selective amplifier. (The length of the string for C,,, was about I m ; the amplitude of the fundamental was very much less than that of the first overtone.) 4. CONCLUSION
The instrument has been shown to have the required accuracy for tuning purposes, but a small piano tuned exactly to the equal temperament scale was found to appear to be sharp at the bass end and flat at the treble end. This effect is undoubtedly due to the inharmonicity of the overtones, and so work on the measurement of the inharmonicity of some strings in a number of pianos is now being carried out. It is believed by the author that this will throw some light on the amount of deviation from the equal temperament scale necessary to achieve a musically satisfactory tuning. ACKNOWLEDGMENTS The author is indebted to Professor G. D. Sims, Head of the Electronics Department, Southampton University, for his encouragement, and for the facilities provided. He is also indebted to many colleagues for helpful discussion, and, in particular, to Mr. K. G. Nichols for the design of the logic circuits of the single pulse selector. The author acknowledges the co-operation of Professor P. A. Evans, Head of the University Music Department, and his colleagues. Finally the author wishes to thank Mr. B. G. Feasey, a former student of the Electronics Department, who did much of the original design as his final year project. REFERENCES I. R. W. YOUNG1954 Acustica, 4, 259. Inharmonicity of piano strings. E. DONNELL BLACKHAM 1965 Scient. Am. 213,88. The physics of the piano. 3. W. R. KNOWLES,S. W. PUNNETT and E. V. VERNON (to be published) An improved tunable audio frequency amplifier. 4. R. E. KIRK 19593. acoust. sot. Am. 31, 1644. 2.