Physics of the Theremin - Semantic Scholar · We discuss some of the interesting physics behind the design and operation of the Theremin electronic musical instrument. To complement

Physics of the ThereminKenneth D. Skeldon, Lindsay M. Reid, Viviene McInally, Brendan Dougan,and Craig FultonDepartment of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, Scotland

~Received 15 May 1998; accepted 12 June 1998!

We discuss some of the interesting physics behind the design and operation of the Thereminelectronic musical instrument. To complement the theory we present details of a parallel effort toconstruct two versions of this remarkable instrument. One of these designs remains fairly faithful tothe traditional beat frequency oscillator approach that first inspired Theremin’s invention, while theother contains more modern electronics that helps make more reliable the setting up and use of theinstrument. The emphasis on the physical principles continues into a comparison of the twoinstruments. Following completion, the Theremins have been extensively used in public and schoolsscience exhibitions and in lecture demonstrations. ©1998 American Association of Physics Teachers.

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I. INTRODUCTION

In 1921 the Russian physicist Leon Theremin gavefirst public demonstration of his musical instrument that hsince become known simply as the Theremin.1 Theremindied in November 1993, but his instrument lives on throuthe occasional concert performance or soundtrack use.2 TheTheremin is unique in that it is played without there beiany physical contact whatsoever between the performerthe instrument. The pitch of the instrument is controlledthe proximity of the player’s hand to an antenna~mountedvertically in the original design!. A second antenna~tradi-tionally mounted horizontally! senses the proximity of theplayer’s other hand and controls the volume of the tone.

The experience of playing the device is quite differefrom that of performing music with most other types of istrument. A seasoned performer is able to associate poinspace with given notes, rather like having the ability to sepiano keyboard that is invisible to everyone else. For bexperienced and beginner players, the playing techniquakin to a continuous negative-feedback mechanism; thesition of the player’s hands defines an auditory input whthe brain then processes, whereupon the positions ofplayer’s hands change in order that the desired note caconverged upon and sustained. Expert Theremin plahave been few and far between3,4 with only a few namesreflected upon when the instrument is mentioned. Perhnot surprisingly, even the renaissance of the instrumenpopular music circles over the past few years has uncovno real original style or talent to rival the virtuoso performethat there have been.5

In this article we will discuss some of the physics behithe design and operation of the Theremin. The instrumencommonly regarded as rather difficult to construct well, aindeed many enthusiasts believe that nobody has been absurpass the quality of operation of those early vacuum tinstruments built by Theremin himself and subsequencommercially produced under patent. However, we cantempt to use some basic physics in order to overcome sof the common design problems, and to help with this,will present two successfully built versions of the instrumephotographs of which are shown in Fig. 1. While our articwould be incomplete if we did not offer these designsstimulate further experiment, we do not want to lose sighthe physics here. Thus we will avoid communicating lengtechnical detail, leaving that instead to personal requestswhich the principal author will be pleased to assist.

945 Am. J. Phys.66 ~11!, November 1998

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II. PHYSICS OF THE THEREMIN ACTION

At the heart of the original Theremins and indeed mostthe subsequent designs, there are one or more beat-frequoscillators ~BFOs!. These are used either directly or indrectly to generate the tone of the instrument and controlvolume. There are many publications detailing sudesigns6,7 but for now we will concentrate on the physicaspects of the BFO action. Let us first consider the prodtion of the tone with a BFO containing two oscillators geerating independent signals at angular frequenciesv1 andv2 . The basic concept is shown schematically in Fig. 2.

A. The beat-frequency oscillator

The outputs of the two oscillators may be expressed mematically asA1 sinv1t and A2 sinv2t. The BFO action isthat of a heterodyne mixer which multiplies these two signtogether providing an outputVout given by

Vout5A1A2 sin v1t sin v2t, ~1!

which can be expanded using a standard trigonometric idtity to yield

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whereA5A1A2 . Thus the output signal of the BFO has twfrequency components; one at the difference of the invidual oscillator frequencies and another at the sum ofoscillator frequencies. By using a low-pass filter, the sigat the sum of the oscillator frequencies can be almost c

Fig. 1. Photographs of the completed Theremins. The instrument on theis based on an analog design quite similar in theory to Theremin’s origdesigns. The instrument on the right contains more sophisticated electrincluding many digital ICs, although the fundamental physical principthat form the player’s interaction with the device remain the same.

945© 1998 American Association of Physics Teachers

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The basic Theremin action is to alter very slightly the frquency of one of the two oscillators so that the beat fquency in Eq.~3! changes also. In practice this is achievby building one of the oscillators in such a way that tcapacitive effect on the pitch antenna by the player’s hslightly tunes the oscillator. Now, the presence of the pler’s hand is not a very significant effect, resulting in a cpacitance change of order a few picofarads, as we shalllater. However, if the oscillator frequency is arranged tovery high, say around 1 MHz, then we require only a fquency change of 1:1000 to give an offset in the beatquency of 1 kHz. This is a significant fraction of the musicscale, being over four musical octaves above the lothreshold of the audio hearing range~around 50 Hz!.

Let us pause for a moment to clarify some of the physin what is going on in the above signal mixing process. Iimportant to realise that the above mixing is not the sameadding the two signals, perhaps an obvious enough obsetion at first. However, adding the signals will also lead to tproduction of beats, as can be found in most introductphysics texts under superpositions of waves, and as poiout to us by many students who were puzzled when shothat simply adding the two oscillator signals producedtone in a speaker. The point to realise, which perhaps ogoes unemphasized but which comes to light immediawith experiment, is the following: When adding~superpos-ing! two waves of slightly different frequency the beats prduced are amplitude beats, modulating only the power ofcombined waveform. There is no actual power at the bfrequency in this case. However, in taking the product of twaves, you do actually channel power into new frequencnamely the sum frequency and difference~beat! frequency.Another way to look at it is in terms of a Fourier expansiwhich, for a given signal, indicates the distribution of amptude or power over frequency. In the case of a superpositthe Fourier expansion is already determined by the termsrepresent the signals you start with, whereas in the casour product here, it is the terms in Eq.~2! that form theFourier components. We could mention in passing thatsuperposition idea can work to produce real signal powethe difference frequency, but only if we introduce some sof nonlinearity into the adding process. For example, a sigproportional to the square ofA1 sinv1t1A2 sinv2t wouldgenerate cross terms that can be reduced to identify a wwith frequency (v12v2)/2. In practice such nonlinearitie

Fig. 2. Simple schematic of a BFO used as a building block in Theredesign.

946 Am. J. Phys., Vol. 66, No. 11, November 1998

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can be introduced simply by a semiconductor diode.course, these are the principles employed in the early rademodulation circuits which relied upon the nonlinear proerties of ‘‘cat-whisker’’ junction diodes to channel signfrom a high carrier frequency down to~audio! sideband fre-quencies.

We should take stock of some of these important prciples of physics that turn up in radio theory, and it is nsurprising that we should come to them so rapidly here. Aall, Theremin originally had the idea for his instrument whworking on short-wave radio equipment and capacitive seing apparatus during the Civil War in Russia. Fortunately,was encouraged to work on his instrument, achievingknowledgement for his first musical demonstrations atEighth Electrical Engineering Congress in Moscow in 192Over the following years, much advancement in radiosearch, particularly in Russia, was motivated by darker mtary applications,8 but the instrument had already been sufciently well introduced to the world, and three companincluding RCA had engaged in its mass production bylate 1920s.

Returning to Eq.~3!, we could state that the clevernessTheremin’s idea is summed up in the discussion followithis equation, that the small capacitive effect of the playehand is up-converted to a perceptively significant changeaudio frequency. The practical issues perhaps become mevident; we seek to build two independent oscillators onewhich is very stable, the other being linked to the pitch atenna. The nominal frequency of each should be high, ctainly radio frequency, and identical when no player is nethe instrument, whereupon the fractionally small changeantenna capacitance introduced by a player’s hand transto a significant change in the beat frequency. In a wedesigned instrument the beat frequency can be increasesentially from zero when no player is near to many kilohewhen the player is almost touching the antenna. We will ndiscuss some aspects of the capacitive effect on the antthat is so crucial to achieving a good Theremin action.

B. Capacitive influence of the player

Before we can investigate the effect of the player onantenna capacitance, we first must appreciate the effect oantenna capacitance on the actual oscillator frequency,hence the Theremin pitch. In order to do this, it is betterhave a design to make reference to, so now we will introdour first version of the instrument: the ‘‘traditional’’ analomachine. The circuit schematic is shown in Fig. 3. The scmatic includes the electronics for both the pitch and volucontrol, but for now we are interested only in the pitch ccuit. We have sectioned the schematic in Fig. 3 to highlithe important circuit blocks. We need to consider the blotitled pitch oscillator 1, for it is the frequency of this ocillatothat depends on the antenna capacitance. Although the cilooks complicated, the basic physics can be extracted fonly a few components located around the pitch antennathe left-hand side of the diagram. For clarity we will makreference to Fig. 4 which shows the basic oscillator in greadetail. This is a Colpitts ocillator which uses off-the-shecomponents and can be more readily made than many opopular designs, such as the Hartley circuit which requirespecially tapped inductor. The oscillation frequency is givby

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946Skeldonet al.

Fig. 3. The complete analog Theremin circuit showing~a! the pitch control circuitry and~b! the volume control circuitry.

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whereC2 can be regarded as a parallel combination ofcircuit component valueC, the antenna capacitanceCA , andall other stray capacitances between the circuit and thetennaCS . It is the value ofCA that our Theremin player willadjust. The instrument is initially set up so thatC2 takes avalue that causesf 0 to match the resonant frequency of pitcoscillator 2 in Fig. 3 when the player is at a suitable initdistancex0 from the pitch antenna; we will denote the valuof antenna capacitance for this case asCA(x0). When the


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player’s hand is a closer distancex from the antenna, we willdenote the modified capacitance asCA(x). In view of Eq.~4!and the fact thatC2(x)5C1CS1CA(x), we can then writethe frequency of the Theremin tonef T as

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Now we need to consider what antenna capacitancemight expect for a typical design. In our analog Theremshown in Fig. 1, the pitch antenna is mounted as a vertrod, around 0.5 m high. The capacitance of such an ante

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can be deduced from potential considerations regardinglindrical lines of charge in space. However, invariably thecalculations do not consider the complex effect of thetenna being near earth and it is more appropriate to citesults from the field of radio engineering where formulae exfor the capacitance of vertical antennae in realistic confirations. For example, after converting the expressions foin Terman9 to SI units, we can obtain the capacitance of tantenna in the absence of any player, which we will denby CA(`), using the formula

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where h and d denote the height and diameter of the rantenna respectively,e0 is the permittivity of free space(8.85310212 F m21), andk is a constant depending on hofar above the ground the antenna is mounted~k;0.4 for anantenna mounted almost at ground level!. Taking the valuesh50.5 m, d50.01 m, andk50.4 we obtainCA(`)'7 pF.We can neglect the stray capacitance for now since itmost likely be small compared withC1 in the oscillator cir-cuit ~100 pF in Fig. 3!.

Next we should investigate how much change in antecapacitance might be expected when a player brings hiher hand near the instrument. One approach would bconsider the player’s hand as a ground plane~of infinite ex-tent! approaching the antenna. This will give a many-timoverestimate but at least allows us to make some initial heway into the problem. In Fig. 5 the presence of the playehand ~ground plane! can be modelled as the addition of aimage antenna whereupon the extra capacitance can bduced from standard results for a twin-cylindricconductor.10 The extra capacitanceDCA is given by

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where it is assumed that the player proximityx is somewhatlarger thand. We have included a reduction factor of 10Eq. ~7! to balance the overestimate that the formula wootherwise give. We justify this in terms of further modellin

Fig. 4. Diagram of a transistor oscillator following the Colpitts desighighlighting the components that determine the resonance frequencyhence affect the Theremin pitch.


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the player’s hand and arm rather like a horizontal strip alothe ground plane of Fig. 5 with a width typically aroundtenth of the antenna height. Since the contributions tocapacitance decay with distance, we can neglect the cortion caused by the arm not being infinitely long, and takeantenna height to be the reference rather than the infinihigh ground plane in judging the correction factor. To obtasome idea of the magnitude of the effect, we can calcuthat a player holding his or her arm 10 cm from the verticantenna produces an additional capacitance of only 0.4

The modified antenna capacitanceCA(x) is thus given byCA(`)1DCA , and with the help of Eq.~5! we can nowderive a relationship between the Theremin pitch andplayer’s proximity to the antenna. It is instructive to viethis dependency graphically for the parameter valuesevant to the instrument described here.

We have done this in the plot of Fig. 6 which shows tmusical ‘‘air-key-range’’ of the instrument vs playing proximity for an instrument that has been initially tuned so thf T is nearly zero for a player standing 1 m from the antennaNote that at this distance it is the player’s body that forthe dominant capacitive effect, and therefore we cannot tthe lower bound in Eq.~5! to beCA(`) as perhaps might afirst be thought.

Figure 6 shows that the Theremin’s musical key patternquite linear over a reasonable range but deviates fromlinear behaviour when the player is very near to the ante~the linear behaviour is also deviated from when the playevery far from the antenna although this is not so obviofrom Fig. 6!. The behaviour shown in Fig. 6 agreed rathwell with experimental data gathered from the instrumeitself. The musical transfer function suggested by Fig. 6realised when trying to play the instrument, for it is alwaeasier to perform in a key transpose which occupies the ctre of the linear range. If, during a performance, the plabegins to veer toward the antenna, then the movementfingers, let alone the entire hand, cause rather large p

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Fig. 5. A first approximation to the effect of a player’s hand on the capatance of the antenna can be deduced if we extrapolate the vertical handground plane and model the result using an image antenna.

948Skeldonet al.

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excursions. A characteristic of the Theremin playing stylethe ease with which a player can induce vibrato on the toIn view of Fig. 6 it is not surprising that a small tremor of thhand is enough to produce this effect, which has becomhallmark of the Theremin sound.

C. Lock-in effect and low frequency performance

In this section we will show how some basic physics chelp draw attention to a common problem with Theremlow frequency performance, ahead of any construction.effect that must be suppressed is the tendency for twooscillating systems of similar frequency to synchronisegether and run at a common frequency. In the Theremin,would cause a problem if the locking happened at a bfrequency above the hearing threshold, causing a suddensation in tone, rather than a smooth decrease in pitch wthe player moves further from the antenna.

The locking phenomenon is seen in various physical stems. For example, in a ring-laser-gyroscope, a signal pportional to the absolute rotation of the device is derivfrom the difference in mode frequencies of two countrotating laser beams in an optical ring cavity.11 In this case,the scattering of light at the mirrors causes a small quanof light to be exchanged between the two modes, reducperformance at small rotation speeds where the mode

Fig. 6. The relationship between the Theremin tone frequency and the pimity of the player’s hand to the pitch antenna, and how this relates toeffective musical key range for the instrument.


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quencies become similar. On the other hand, the effeclock-in is used to advantage in high power Nd:YAG lassystems, where a high power but noisy laser cavity is locto a quieter more stable low power laser through the injtion of some of the stable light into the high power lascavity.12 What we want to emphasize here is that the concof locking is very much a general principle of physics, athough the precise coupling mechanisms for a given situacan be quite different.

In the case of the Theremin BFOs, we might have capative or inductive cross talk between the oscillator circuits,ground currents and power supply ripple to contend wThe diagram in Fig. 7 shows qualitative linewidth curves fthe Colpitts oscillators in Fig. 3. The locking effect is mamore problematic when the linewidth is wide compared wthe frequency offset from the other oscillator, as is the cfor the fairly low-Q circuit of Fig. 4. However, some simplesteps can be taken to reduce the problem. First, electromnetic cross talk can be suppressed significantly by separascreening each oscillator. Ground currents have toavoided as much as possible by the use of a good comground mecca to which all oscillators are connected. Posupply decoupling must be included which can take the foof series inductors with capacitors to ground at the powentry points of each oscillator; this will attenuate any hifrequency ripple on the dc supply lines.13 Without these mea-sures, the Theremin circuit of Fig. 3 would lock at a fehundred hertz, seriously affecting the useful range ofinstrument. In Fig. 3 the 100-V resistor and 47-mF capacitorto ground in each oscillator form a decoupling filter, whethe components are connected at the power supply epoints to each oscillator. In addition, series inductors inpower line outside each oscillator provide additional decpling ~not shown in Fig. 3!. With the system carefully decoupled, modularised, and with each block screenedshown in Fig. 8 we managed to push the locking effect dobelow 50 Hz so that the player is unaware of it.

D. Tuning and playing the analog Theremin

The broad linewidth of the oscillators in an analog Theemin design permits a large drift of running frequency,fluenced, for example, by thermal drifts in component vues. The instrument can often require tuning when switcon and possibly fine adjustment thereafter. In order to avthe inconvenience of having to open the cabinet and trimoscillator frequency every time the instrument requiredjusting, we included a novel electromechanical tuning cotrol. The control shown in Fig. 9 consists of a simple dicapacitor which appears in parallel with the antenna. T

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Fig. 7. Coupling mechanisms, which may be of relatively high order in the system, can lead to the synchronisation of the two free running oscilla
949Skeldonet al.

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Fig. 8. Block diagram and photograph illustrating how the various circuit modules of the analog Theremin were individually screened and interconneground originates from the power supply and is used to ground the chassis of each fixed oscillator box~i.e., the oscillators not connected to the antenna!.Subsequent ground lines are made via the shielded cable carrying the signals to other boxes.

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disk spacing is controlled by a long plastic shaft whichterminated via a threaded collar on the left-hand side ofcabinet. The long shaft was included so that the playerturn the tuning control without confusing the adjustmentcoming too close to the pitch antenna. By turning the conknob, the spacing between the plates is finely adjusted,extra capacitance given by

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where we are considering only air as the dielectric. NotthatCS could be regarded as a changing stray capacitancthe previous discussion. We implemented such a control wr 52 cm and 0,a,1 cm, giving the ability to trim the ca-pacitance to any value above;1 pF. During initial setup,this control is set toa;2 mm, givingCS;6 pF. This givesa wide range of control to correct for drifts of both polaritiin the BFO output.

Fig. 9. Mechanical means to adjust the capacitance between each anand ground, illustrated here for the pitch antenna. Initial coarse tuninachieved using the trimmer capacitors in each fixed oscillator, but tsubsequent adjustments can be made without the need to reopen theemin case.


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Now that tuning the instrument is relatively straightfoward, there is nothing to prevent practising and playing msic with the Theremin. However, we should note that anotpotential problem is the detection of radio signals, andpickup of other electromagnetic fields, by the antennae ofinstrument; we observed this phenomenon for some timeour own machine before retuning the oscillators. It maywise to research the frequencies of local radio stations bedeciding on the frequencies for the BFO oscillators. Hoever, we have found that performing the instrument inarea of poor radio reception and away from fluorescent liging is the best safeguard against such interference.

Finally, we reflect upon the fact that we have not yet dcussed any method of controlling the volume of the Themin tone, even though we have introduced a circuitdoing so in Fig. 3~b!. It is fitting now to introduce a morestable design of instrument which addresses many of thetors we have discussed above, and this is doubly convenin that our new design is best introduced via the discussof the volume control technique.

III. BUILDING BLOCKS OF A MODERNTHEREMIN

As with practically any instrument, the facility to adjusvolume adds a great dimension of expressiveness to thesical sound produced. This facility is particularly importain the Theremin since without it, the transition from one noto another would necessarily always be a portamentopitch-bending effect. We will now discuss the volume cotrol circuit for our analog Theremin as this lends itself weto introducing the conceptual building block for a more avanced design; the concept is that of a proximity-to-voltaconverter.

A. Volume control

Consider again the analog Theremin circuit shown in F3. The volume circuit is shown in Fig. 3~b! and contains thesame BFO design as the pitch control circuit in Fig. 3~a!. Theoutput of the volume mixer is passed through a low-pfilter with a corner frequency of about 1 kHz. This produca signal with frequency and, more importantly here, amptude that varies in response to perturbations in the antecapacitance. This filtered signal is amplified and rectified a

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950Skeldonet al.

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Fig. 10. The various stages of the digital control module whose function is to generate a dc voltage that depends on the proximity of the playersthe antenna. This circuit block will be referred to as a proximity detector~PD! in the text.
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passed through a simple integrating filter which providedc control signal. This dc signal is a direct function of tfiltered signal amplitude which in turn depends on the potion of the player’s hand. The integrator time constant mbe chosen so that fluctuations in the dc level can responthe fastest movement of the player’s hand and is taken toabout 0.1 s here. Once a control signal has been obtafrom the BFO in this way, it can be used to control a wivariety of electronic devices. In the circuit of Fig. 3~b! the dccontrol signal is processed so that it changes from 0 V~closeproximity to antenna! to 24 V ~far away from antenna!. Thisconstitutes a convenient control voltage range to apply tJFET in voltage-controlled resistor~VCR! mode. The JFETand the 4.7-kV resistor form a potential divider across whicthe audio tone from the pitch circuit is connected. The outis taken from the middle and ranges in amplitude frommost full input signal when the JFET resistance is high~24V applied to its gate! to practically no signal when the JFEresistance is low~0 V applied to its gate!. This configurationallows the volume to be attenuated by moving closer toantenna, as in the early RCA units. We measured a 40suppression in volume using this simple design. Howeverthe problems of lock-in and drifts in tuning apply also to tvolume BFO. Therefore we will now introduce a more moern approach motivated by the idea of control-voltage pduction described above.

B. Proximity-to-voltage converter

A key feature of the analog Theremin design discussedfar is that the output sound is directly related to the charteristics of the high frequency signals generated by epitch oscillator.14 Some argue that this is what gave the orinal Theremins their distinctive sound, with distortion anonlinearities in the vacuum tubes all contributing to toutput waveform. However we may not wish to carry thfeature here; therefore one aim of our modified circuit is tit distances the final output sound from the waveforms pduced by the BFO blocks. Instead, the modified equivalenthe analog BFO is used merely to generate a dc control v


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age ~CV!, rather like in the volume control circuit of Fig3~b!. These CVs can then be used to feed into a wide varof devices including voltage-controlled oscillators~VCOs!,voltage-controlled amplifiers~VCAs!, and possibly evenvoltage-controlled filters~VCFs!. In addition it is a simpleextension of our design to produce a CV with suitable ranto pole the CV gates of analog synthesizers. We couldthe functional block that generates each CV by the nameproximity-to-voltage converter or proximity detector~PD!.In the simplest Theremin, we would require only one PD aone VCO to generate music. To emulate the properties‘‘traditional’’ machine, we would require two PDs, one feeing into a VCO for pitch control and the other controllingVCA for volume adjustment.

A schematic of the PD unit that we have built is shownFig. 10. The output of each digital oscillator is a square wawith frequency set by the values ofR and C. Two suchoscillators are used in the PD, one of which is connectedthe antenna. The outputs of these oscillators can be cbined by feeding them via buffers into an exclusive-OR~X-OR! gate. By considering the truth table for this gate, givin Fig. 10, it becomes apparent after some careful thouthat the output will be a signal whose rising edges ocroughly at the sum of the input frequencies, but whose dcycle varies at the difference of the input frequencies. Thefore by integrating the output of the X-OR gate with a suable low-pass filter, it is possible to recover a signal whoprimary Fourier component is at the difference frequenThe signal is fed through a Schmitt trigger to producesquare wave, which is then passed to a monostable trigwhich produces a series of fixed-width pulses at the diffence frequency. Finally, these are integrated with a seclow-pass filter to yield the output CV. Initial frequencmatching between each oscillator is achieved by the usthe variableR. Since there are no inductors and a minimnumber of capacitors, we can employ precision resistorsa stable type ofC in order to achieve much less drift in thidesign.

Having introduced the basic PD building block, the appcations are best left to the imagination of individual expe

951Skeldonet al.

Fig. 11. The complete digital Theremin circuit showing~a! the pitch control electronics and~b! the volume control electronics.

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menters; however for completeness we will present our odesign for a digital Theremin with pitch and volume controlers.

C. Digital Theremin circuit

The schematic is shown in Fig. 11 with the pitch avolume control blocks shown in Figs. 11~a! and 11~b!, re-


nspectively. We will refer to the pitch circuit first of all. Weused the ICL8038 VCO from Harris Semiconductors.15 Thisdevice produces a frequency sweep of 1000:1 when its ctrol voltage input is varied from about two-thirds of its suply voltage to a few tenths of a volt more than its positisupply voltage. The necessary control voltage is produceda PD of the type discussed above together with approprCV processing using a simple operational amplifier circuit

952Skeldonet al.

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perform relevant algebra. After initial tuning, the basic Pgenerated a CV of 0 to11 V as the antenna was approacheSubsequent adjustments could be made using the 10 andkV variable resistors around the op-amp following the PThese can correct for any drift between the two oscillatorsthe PD and effectively replace the tuning control of the alog design. In fact it was found that only the 10-kV resistorneeded adjustment and so it appears on the front panel, afrom the pitch antenna as possible. We found that we coachieve a sweeping of the 8038 frequency from aroundHz to over 10 kHz over a playing range of around 0.6m wthe circuit of Fig. 11~a!.

The relationship between playing position or proximitypitch antenna and the frequency of tone produced is diffito predict for the digital device, because it will involve thtransfer function of PD1 as well as the VCO in Fig. 11~a!.Therefore, in order to produce a diagram equivalent to Figfor the digital instrument, we experimentally measuredtransfer function from playing position through to output frquency; the results are shown in Fig. 12. It can be seenour digital Theremin preserves a constant octave intervato a much higher frequency than does the analog desHowever, the very low frequency behaviour was not as goas that seen with the analog Theremin and this perhapsflects the extra effort we put into decoupling in the anacase to avoid lock-in and achieve the sub-100 Hz permance.

Returning to Fig. 11~b! it can be seen that the volumcircuit is identical to that for the pitch control, up to the enof the PD block. The extra inverter stage after PD2 wfound to be necessary for purely technical reasons stemmfrom the requirement that touching the volume antenshould produce zero volume. Again, the 10-kV offset controlis mounted on the case, this time far from the volumetenna. The VCA chosen here is based on the SSM-2018sen for its relatively low cost and robust performance16

Fig. 12. The relationship between the digital Theremin tone frequencythe proximity of the player’s hand to the pitch antenna, and how this relto the effective musical key range for the instrument. It is interestingnotice that the octave interval remains fairly constant up to much higfrequency than for the analog Theremin.


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Maximum and minimum gain for this VCA correspondspin 11 being brought, respectively, negative and positivea few tenths of a volt with respect to ground. The outpfrom the 8038 VCO can only drive rather high impedaninputs, and this accounts for the unusually high choice o100-kV potentiometer used to attenuate the signal at theput of the VCA. In normal operation this potentiometshould be set so that when the VCA has maximum gain,clipping of the signal takes place. However in practice it wdecided to allow the input signal to be sufficiently high soto result in clipping of the output signal for the maximugain of the VCA chip. This results in a square wave at toutput when a sinusoidal wave is present at the input,thus effectively generates new harmonics in the sound. Thharmonics are suppressed when the gain of the VCA isduced, whereupon a sinusoidal wave as pure as the iwaveform is produced. Further reduction of the VCA garesults in a normal decrease in volume of this sinusoiwave. We found it possible to produce a volume sweepover 80 dB with the SSM-2018 IC resulting in a range thbegins at inaudibility~when the player touches the antenn!and ends with around 20-dB amplification of the input sign~when the player is distant from the antenna!.

D. Comparison of the analog and digital Theremin

Finally, we may compare the relative attributes of the alog and digital designs. From a teaching-aid point of viethe analog Theremin is better suited for conveying the phics of the instrument. There is a clear overlap with radtheory and associated principles, as well as the otheramples of applied physics outlined in Sec. II. Neverthelethe analog Theremin of the basic design in Fig. 3 does sufrom disadvantages of component drift and susceptibilityexternal interference. In the digital design, we sought to pduce an easier to operate instrument, using a control-volapproach borrowed in essence from the analog circuit’s vume section. In doing so, we found the digital Thereminoffer a more linear octave range up to higher frequencythe expense of reduced range at very low frequency.degree of linearity of the musical ‘‘air-keyboard’’ can bdeduced from the plots of Figs. 6 and 12. Perhaps a cleway of comparing these plots is to graph the logarithmfrequency for each instrument, since a truly linear keyborange will be reflected in this graph as a straight line. Whave done this for the playing range up to 0.6 m in Fig. 1An even clearer way to visualize the implications for playimusic is shown in Fig. 14 where we have shown the efftive key ranges over a 0.6-m playing range for each insment. The top key range represents five octaves of a norpiano range, and the relative distortions are shown foranalog and digital Theremins underneath. Our experiehas been that beginner players find the analog deviceeasiest to play, probably because there is a larger ranglinearity all the way out to 1 m from the antenna, makingeasier to converge on a given note. However, once a feefor the playing style has been established, the more linupper octaves of the digital Theremin can be more skillfuused.

The volume control in both instruments felt to have qusimilar transfer functions from playing position to attenution of volume, although we did not make any quantitatimeasurements of this. However, the option of settingmaximum volume of the digital device so that the signclips at the VCA output allows more expressive playing st

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to be achieved. Lastly, as perhaps might be noticeable four photographs in Fig. 1, we tried to forge a stylistic diffeence between the analog instrument and the digital devendeavouring in particular to keep the analog instrumquite similar to the look and feel of the commercially prduced Theremins of the 1920s.

Fig. 13. Plots of the logarithm of frequency~equivalent to a piano keyrange! against proximity to the pitch antenna of each instrument. Musiclinear playing ranges are indicated with the arrow intervals. These intercan be projected onto the vertical keyboard to show immediately the mukey range over which each instrument offers good linear behaviour.course, the whole musically useful playing range for each instrumenmuch wider than these linear intervals, but skillful playing requires mupractice in order to become familiar with the entire transfer function of einstrument.

Fig. 14. A qualitative illustration showing the distorted key ranges foranalog Theremin~middle! and digital Theremin~bottom! compared with theplaying range that an ideal linear keyboard of roughly the same numbeoctaves might provide~top!. This picture clearly shows the merits of thanalog instrument at lower frequencies while also highlighting the advtages of the digital design over the higher octaves. Note that unlike prevdiagrams, the proximity decreases to the right here in order to enablekeyboards to be drawn intuitively.


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IV. CONCLUSION

We have outlined the physical concepts behind the desand operation of the Theremin, making reference to an alog design and taking the opportunity to present a morenovative digital device. The overall number of octaves theach instrument could cover was approximately the sahowever the analog Theremin possessed better lowquency performance, concentrating the upper octaves innarrow playing range. The digital instrument had a failinear playing range extending over three octaves abmiddle A. The volume circuit was arranged in both instrments to null the sound level when the player’s hand wtouching the antennae.

The digital circuit has much scope for expansion usexternal control sources to affect the Theremin sound.example, adding the output of a drum machine to thecontrol level for the VCA~through a resistor into the virtuaground of the inverting adder! allows the sound level to bepulsed to the beat. The pitch can be modulated in a simfashion by adding in a signal through a resistor to the virtground of its inverting adder stage. If variable sinusoidsignals are applied to these same points, then the rewould be an adjustable tremolo and vibrato. We haachieved an interesting variety of sounds using a combtion of these techniques. Another possibility is the controlmodern digital synthesizers via the addition of a commercCV to MIDI converter using the PD to generate the CV. Bemploying a third antenna and another PD, a VCF or renant filter block could be included to produce even mointeresting sounds; however we leave the implementationsuch upgrades and expansions to enthusiastic experimeelsewhere.

We should not forget that the Theremin is first and fomost the musical instrument that stimulated the birth of msic using electricity. Although it is strictly speaking an eletronic musical instrument, we can still regard it in manways as being more closely related to a ‘‘passive’’ instment than is say, the electronic synthesizer. This is duepart to the simplicity of the tone produced, but we might aargue that it is because so many aspects of its operationstill be described by principles of applied physics, rather thof electronics which can so often appear more of an art ta science. Indeed, some of the early analog Theremins wunique in providing a tone that was closer to a pure swave than almost any other acoustic instrument you comention. Its sound is remarkably distinctive, and in osense, much more characteristic than the sounds fromday’s digital synthesizers which are sold largely on thability to achieve very complex timbres or realistic instrment sounds. The effect of adding harmonics to the baTheremin-style sound is well demonstrated by our digdesign, where the signal levels from the amplifier canmade to clip. There is teaching value to this demonstratwhich could be related to the role of overtones in definithe acoustic timbre of other types of musical instrument.addition, many other aspects pertaining to the principlesoperation have teaching value, and as we mentioned eathe simplicity of the analog instrument affords its strengththis area.

Finally, we acknowledge that an interest in Theremininstrument would not be complete without due consideratof the remarkable events in the life of the physicist himseHis story is one of the most amazing scientific biographiesthe twentieth century, perhaps because it enrobes an en

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ing passion in science and society. This passion contindespite the interval of a generation when Theremin wasvoluntarily isolated, not only from the people he knew aloved, but also from the research field toward which he hchosen to direct his knowledge of physics.17

ACKNOWLEDGMENTS

We would like to thank A. I. Grant, J. Hough, H. Ward,Barbour, the Honours Laboratory Staff, and the GravitatioWaves Group in our department for their help and advthroughout the circuit construction phases. Financial supwas provided by the University of Glasgow. KDS was suported by United Kingdom Particle Physics and AstronoResearch Council~PPARC! and British Petroleum/Royal Society of Edinburgh~BP/RSE! fellowship awards. The otheauthors were all partly supported by student support gra

1G. Anfilov, Physics and Music~MIR, Moscow, 1966!. This rarely citedbook was translated from Russian by B. Kuznetsov and describes Temin’s early research in Moscow, although it does not describe the aming course that Theremin’s life took after 1938, when he vanished withtrace from his New York apartments.

2Since Theremin’s death, there has been an ever-increasing band of pomusicians that seem to have just ‘‘rediscovered’’ his instrument’s valua source of ‘‘cult sound.’’ However, the Theremin has been used osionally in popular music for many decades; one famous example isBeach Boys hit ‘‘Good Vibrations’’ where the Theremin and cello provibacking melody.

3T. Rhea, ‘‘Recordings; Clara Rockmore: The Art of the ThereminComp. Music J.13, 61–63~1989!.

4R. Moog, ‘‘Theremin Virtuoso Clara Rockmore—Recollections of Gnius,’’ Keyboard Magazine~February 1994!.

5In the early twentieth century there were around seven-hundred prosional Thereminists registered with the musicians’ trade union. It shoalso be noted that Leon Theremin’s great niece, Lydia Kavina, is ancomplished present-day Thereminist.

6R. Moog, ‘‘A Transistorized Theremin,’’Electronics World ~January1961!.

7R. Moog, ‘‘A Transistorized Theremin,’’Elementary Electronics~May/June 1972!. Of course, this and the previous reference are only two


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amples from a multitude of Theremin designs spanning many decadcomprehensive list may be found using Ref. 17.

8Theremin’s disappearance from his New York apartments went unplained for over 50 years. His wife and most of his closest friends psumed him dead. However in 1991, research for Steven Martin’s vidocumentaryTheremin: An Electronic Odyssey~Orion Classics Release1993! led to a remarkable chance encounter with the physicist, then a95, in Moscow. In the film, Theremin explains how he was abductedthe forerunner of the KGB, an act motivated by the need in Russia attime for expertise to expediate technological military developments.Ther-emin: An Electronic Odysseywas premiered on UK’s Channel 4 Televsion. Just one day after the thought-provoking film was shown, Leon Temin’s death was announced in Moscow.

9F. E. Terman,Radio Engineers Handbook~McGraw–Hill, New York,1943!.

10G. P. Harnwell, Principles of Electricity and Electromagnetism~McGraw–Hill, New York, 1938!.

11K. D. Skeldon, ‘‘A Museum Scale Ring-Laser-Gyroscope to Demonstrthe Sagnac Effect,’’ University of Glasgow Vacation Project~Internal Re-port 1991!.

12A. E. Siegman,Lasers ~University Science Books, Mill Valley, CA,1986!.

13P. Horowitz and W. Hill,The Art of Electronics~Cambridge U.P., Cam-bridge 1990!, 2nd ed.

14The waveforms of each oscillator and the resulting beat waveform pduced by the chosen mixer all have an impact on the sound. In the anTheremin circuit, we used SBL-1 mixers~Mini Circuits Laboratory,RFDesigners Handbook, 1994!. In this implementation, the two oscillators ineach BFO provide the radio frequency~rf! and local oscillator~LO! inputsto the mixer, and the intermediate frequency~IF! output from the mixer~which is nonstandard in being down near dc here! provides the tone. Wearranged the signal levels at the SBL-1 to be just below clipping~around1.2 V! whereupon the output waveform was reasonably sinusoidal abofew hundred hertz, beginning to distort more toward dc. However,found the output sound timbre to be very authentic compared to recordof early Theremins.

15Harris Semiconductors have their main home page at httwww.semi.harris.com and information for the ICL8038 can be obtainusing the data-sheet links found therein.

16Precision Monolithics Inc, Data Book-Analog Integrated Circuits, 199017The Theremin home page can be found at http://www.nashville.

;theremin and contains many sources of reference on Theremin’s lifework, including much on his ‘‘missing years.’’

ENTHUSIASM

As a graduate student at the University of Hawaii,@Heidi Hammel# recalls, she was ‘‘chided’’by the committee reviewing her thesis for her excessive enthusiasm. ‘‘I think the way one guy putit,’’ Ms. Hammel laughs, ‘‘was that I spoke with exclamation marks when I talked. It’s just theway I talk. I can’t help it.’’

‘‘I worry sometimes when I give talks at scientific meetings, because the style is to stand upthere,’’ she adds, her speech growing slower and deeper, ‘‘look very serious, and present yourresults because they’re so very important.

‘‘I don’t care,’’ she says, ‘‘I just get up there and have a good time. I wonder if sometimespeople lose the message of what I’m talking about, because it’s so different from the normalscientific style, which isso boring.’’

Kim A. McDonald, ‘‘The Comet Drama’s Biggest Hit,’’ The Chronicle of Higher Education, 27 July 1994.

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Documents

Physics of the Theremin - Semantic Scholar · We discuss some of the interesting physics behind the design and operation of the Theremin electronic musical instrument. To complement