Was Something Wrongwith Beethoven’sMetronome?Sture Forsén, Harry B. Gray, L. K. Olof Lindgren, and Shirley B. Gray

Mathematicians and scientists seldomrest easy when the numbers are notright. The search for understandingis never far from mind. In many ways,this is why mathematics is so much

fun. Similarly, conductors and musicians, whoimmediately recognize wrong notes, are perplexedby Beethoven’s metronome markings. Some of histempo markings, even on many of his most popularclassics, have been considered so fast as to beimpossible to play. What is the problem? Why?

The pianist and musicologist Peter Stadlen(1910–1996), who devoted many years to studiesof Beethoven’s markings, regarded sixty-six outof a total of 135 important markings as absurdlyfast and thus possibly wrong [1], [2]. Indeed, manyif not most of Beethoven’s markings have beenignored by latter day conductors and recordingartists. This situation is all the more puzzling sinceBeethoven himself was a strong proponent of themetronome and took the instrument to heart whenit first came into his hands about 1816. He evenexpressed satisfaction that finally his intentions as

Sture Forsén is senior professor at Pufendorf Institute forAdvanced Studies, Lund, Sweden. His email address issture.forsen@pi.lu.se.

Harry B. Gray is founding director of the Beckman Insti-tute, California Institute of Technology. His email address ishbgray@caltech.edu.

L. K. Olof Lindgren is scientific secretary at the SwedishFoundation for Strategic Research, Stockholm, Sweden. Hisemail address is olof.lindgren@stratresearch.se.

Shirley B. Gray is professor of mathematics at Califor-nia State University, Los Angeles. Her email address issgray@calstatela.edu.

DOI: http://dx.doi.org/10.1090/noti1044

to the tempi in his more important compositionswould be made generally known.

The literature on the subject is enormous.However, this article will focus on the salientfeatures of the early metronomes as seen throughthe eyes of a scientist, engineer, or mathematician.We investigate their early history, how they wereconstructed, and their mechanical properties. Weinvestigate possible sources of error as to whyBeethoven may not have been able to correctlynote reliable and transferable time measures.We hope to demonstrate that there are possiblemathematical explanations for the “curious” tempomarkings—explanations that hitherto have notbeen considered except perhaps by Stadlen, whoeven went so far as to locate Beethoven’s ownmetronome.

Musical Notation and Tempo Indications inBeethoven’s LifetimeThe musical notation used in European classicalmusic—the five-line staff indicating pitch and notesof different appearance, whole notes, half notes,quarter notes, etc., to indicate the relative lengthof a tone within a time interval such as the “beat”or the “bar”—was essentially in widespread useby the end of the seventeenth century [3]. Whatwas lacking in these early days was a universalway of indicating the tempo. Relative tempi couldbe used. There was the Italian system, still inuse, with expressive words that run the gamutfrom fast to slow, proceeding from prestissimo allthe way through vivace, allegro, to adagio andthe sluggish larghissimo. There was, and is, alsothe German equivalent, again from fast to slow,going from schnell, perhaps preceded with an

1146 Notices of the AMS Volume 60, Number 9

amplifying sehr , towards sehr langsam. Of coursethere is the French as well. The verbal way ofmarking tempi gives considerable freedom to theperforming musician and thus a certain amount ofartistic latitude to adapt to the current moment.Nevertheless, a number of composers, performingartists, and conductors were not satisfied withthe lack of precision. In fact, already in the earlyyears of the seventeenth century, the human pulsewas used for timing. It is reported that the pulsewas taken as eighty beats/minute—which seemssomewhat high but may reflect the level of stressof performing artists. Also, the use of clocks wasproposed by Henry Purcell in the late seventeenthcentury, and then there were devices based on theproperties of simple pendulums.

The study of the period of a simple pendulumis associated with the great Italian scientist Galileo.In Pisa he is thought to have experimented with apiece of string, fixed at one end and with a weightat the other. First, a pendulum is independent ofthe mass of the weight for a constant length of thestring, and secondly, the square of the period oftime varies directly with the length of the string.Galileo’s results were eventually put into use formusical timekeeping. Perhaps the first was theFrench flutist and musicologist Etienne Louliè atthe end of the seventeenth century. His devicewas rather unwieldy—it is reported to have beennearly two meters (approximately six feet) tall andobviously not very mobile. But his basic designwas gradually refined by others who equipped aclockworks device to keep the pendulum swingingand also added a graduated scale by which thelength of the string with the weight could be bothaltered and measured. A drawing of a penduluminstrument, i.e., the metromètre, designed ca.1730 by Count D’Ons-en-Bray is reproduced in anarticle by Thomas Y. Levin [4]. Unwieldy as theymust have been, the pendulum-based timekeepinginstruments seem to have been in use well into thenineteenth century for want of a simpler alternative.Readers may want to compare the metromètrewith Huygens’s-pendulum clock (1673) [5].

But a “quantum jump” in the history of musicaltimekeeping was not far ahead. In Amsterdamin 1812 Dietrich Nikolaus Winkel (1780–1826)was experimenting with pendulums. He made thediscovery that a special variant of a pendulum, inthis case a thin wooden or metal beam some 15–20cm in length and onto which weights could beattached, behaved in an interesting way. If the beamwas able to swing freely around a pivot and twoweights were attached to the beam, one on eitherside of the pivot, the pendulum would beat a steadytime. Furthermore, with proper adjustment of theposition and mass of the weights, it would beateither slow tempi, which previously had required

very long single weight pendulums, or very fasttempi if the weights were moved toward the pivot.The principle of the metronome was born. Readersmay picture the new invention as an augmentedapplication of the Archimedean Law of the Lever.

Winkel must have realized the double pendulumprinciple could be applied to musical timekeeping.But there was obviously more to be done beforethe idea could be transformed into an easy-to-use instrument. Winkel spent a few yearsexperimenting with his ideas. Finally, in August of1815 he described his new invention in the Reportsof the Netherlands Academy of Sciences where italso received extensive praise in a commentary.Though published, Winkel was soon to meet hisnemesis. Winkel was no businessman for he had notpatented his invention. He was soon approachedby an enterprising man from Vienna who wassomething of a ruthless entrepreneur, a mannamed Mälzel.

The Multitalented Herr Mälzel

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Johann Nepo-muk Mälzel(1772–1838) wasquite an interest-ing character. Hewas born in Re-gensburg wherehis father wasa skillful organ

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builder. Young Johannturned out to haveconsiderable musicaltalent. At the age offourteen he was al-ready regarded as oneof the best piano play-ers in the city. Buthe was also a mostgifted mechanical en-gineer. At the age oftwenty he establishedhimself in Vienna andeventually gained ac-cess to a mechanicalworkshop in the fa-mous Stein factory,producer of one ofthe pianoforte instru-ments that Beethovenfavored. Mälzel wasseemingly not withouta well-developed tastefor publicity. Moreover, his talent was in construct-ing musical automatons, devices that appealedgreatly to the public at that time. A recent inves-tigation reveals that he may in fact have copied

October 2013 Notices of the AMS 1147

inventions by others, changing them slightly andthen claiming that they were his own. One of hismost famous mechanical constructions was theremarkable panharmonicon that could imitate allinstruments in a military band—even gun shots! Itwas actuated by a bellow. The notes played weredetermined by pins attached to a large rotatingwooden cylinder, much like in old barrel-organs.The amazing “device” was demonstrated not onlyin Vienna but also in Paris where he sold his firstinstrument in 1807. It caught the eye of LuigiCherubini who wrote a composition named Echofor the panharmonicon. It is indeed sad that theonly copy of the panharmonicon, which for manyyears was kept at the Landesgewerbemuseum inStuttgart, was destroyed in a bomb raid in 1942.

Mälzel’s reputation as somewhat of a mechani-cal wizard grew and in 1808 he received the titleHofkammermaschinist , a title that translates ascourt or royal machinist. He came in contact withBeethoven, who sought help for his increasing hear-ing loss. Mälzel constructed several ear trumpetsfor him, some still on display at the BeethovenHaus in Bonn. During Beethoven’s visit to Mälzel’sworkshop, the problem of musical timekeeping wasalmost certainly discussed, and Mälzel apparentlystarted working on the issue [6]. It appears that

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in 1813 he actually madesome kind of timekeepingdevice, a “chronometer”, pos-sibly based on an earlierdesign by G. E. Stöckler. InJune of the same year Mälzelalso suggested to Beethoventhat he write a symphonyto celebrate the Duke ofWellington’s recent victoryover Napoleon’s troops at Vi-toria in Spain, and he addedthat the composition shouldbe arranged for the panhar-monicon. Beethoven agreedand, as was his habit, out-lined the composition. Thecollaboration ended in a bit-ter conflict between the two.Mälzel considered himselfthe rightful owner of thefinal work although the pan-

harmonicon version was eventually abandonedand the composition rewritten by Beethoven for asymphony orchestra and first performed in thisform in December of 1813. Beethoven dismissedMälzel’s claims and instigated legal action againsthim.

Chess players will be interested to know thatin later life Mälzel purchased a remarkable chessplaying automaton designed in 1770 by Baron

Wolfgang von Kampelen. This automaton wasgenerally known as “The Turk” after the dress ofthe almost life-sized doll that moved the chesspieces. Although seemingly run by an intricatemechanical assembly of cog-wheels and rods,it actually hid a human player in a way thatescaped even the most inquisitive skeptics. Mälzelsuccessfully toured with the Turk, who did beatmost opponents, Napoleon among them, all overEurope. Later he toured the United States where,however, the human involvement inside the Turkwas accidentally exposed. Mälzel, who became awealthy man, died in 1838 of an overdose of alcoholon a ship in the harbor of LaGuaira, Venezuela.The Turk eventually ended its days in 1854 whenit was destroyed by fire at the National Theaterin Philadelphia. Both Wikipedia and the magicianJames Randi have described how the public wasfooled into accepting the idea that no human couldpossibly be hidden inside [7].

But now back to the invention of the doublependulum by Dietrich Winkel in Amsterdam. Newstraveled fast even in the early part of the nineteenthcentury. Mälzel had obtained some informationon Winkel’s invention as early as 1812. It is evenpossible that the two had briefly met that year.But after the publication of Winkel’s invention inAugust of 1815, Mälzel hurried to Amsterdam, metwith Winkel, inspected the new “metronome” andrealized its superiority over his own timekeepingdevices. He offered Winkel money to buy theright to sell the device under his own name. Notsurprisingly, Winkel refused.

Intellectual property rights were rarely enforcedin those days, so Mälzel went back to Vienna, madea copy of Winkel’s instrument, added a gradedscale to the oscillating beam on the side of themovable weight, took the copy to Paris, and sawto it that “his”—Mälzel’s—invention was patentedthere and later also in London and Vienna. He evenset up a small factory, Mälzel & Cie, in Paris forthe production of “his” metronomes.

In 1817 Winkel became fully aware of Mälzel’sactivities and he was understandably upset. Heinstigated proceedings against Mälzel for the obvi-ous theft. The Netherlands Academy of Scienceswas involved as arbitrator and ruled in Winkel’sfavor. To no avail, Winkel had been scooped. Themetronome factory in Paris was up and running.Metronomes soon became very popular. Mälzel wasregarded as the true inventor and the abbreviation“MM” for “Mälzel’s Metronome” was commonlyplaced before metronome measures in printedsheets of music.

1148 Notices of the AMS Volume 60, Number 9

Beethoven and Mälzel’s MetronomeThe first metronomes based on Winkel’s doublependulum principles from Mälzel’s Paris factoryshould have become available in early 1816. It isthus possible that Beethoven was presented witha copy around this time, perhaps as a gesture ofpeace from Mälzel, who must have realized thatBeethoven’s approval of the instrument would begood for business. Not much is mentioned aboutthe metronome in Beethoven’s letters in 1816,perhaps due to the master’s preoccupation withpressing family matters. His brother Caspar Carldied of tuberculosis in November 1815, leavingbehind a nine-year-old son, Karl. Beethoven madeevery effort to become the sole guardian of hisyoung nephew, to the point of taking the caseto court in opposition to Karl’s mother, Johanna.The legal battle went on for more than four years,causing considerable emotional strain on youngKarl [8].

It is reported that in 1816 Beethoven was sopreoccupied by these legal dealings that he stoppedcomposing. But from 1817 onward the metronomewas indeed on his mind. At that time he wroteHofrath von Mosel, “So far as I am myself concerned,I have long purposed giving up those inconsistentterms ‘allegro’, ‘andante’, ‘adagio’, and ‘presto’;and Mälzel’s metronome furnishes us with thebest opportunity of doing so” [9]. Later, his ninthsymphony, in addition to his earlier symphonies,was marked with metronome measures. After areport from a performance of its popularity inBerlin, he wrote to his publisher Schott, “I havereceived letters from Berlin informing me thatthe first performance of the ninth symphonywas received with enthusiastic applause, whichI attribute largely to the metronome markings”[10]. Surprisingly, Beethoven only gave metronomemarkings to one of his piano sonatas, Op. 106,also called the Hammerklavier sonata. Here thefirst movement, an allegro, is marked one hundredthirty-eight beats per minute for the half note,which is extraordinarily fast. The great pianistWilhelm Kempff made some rather harsh commentson this marking in his set of recordings of the laterpiano sonatas, “The erroneous (sic!) metronomemarkings can easily lead to this regal movementbeing robbed of its radiant majesty.”

Assuming Beethoven had one of Mälzel’smetronomes in his possession by 1817, he probablycould not hear the “clicks” of the repetitive beatas he was suffering from approaching deafness atthat time. But, of course, he could see the oscil-lating beam. So what could possibly have causedhim to indicate the fast tempi throughout thatso puzzled subsequent generations of musicians?Have present-day musicians tried to make hiscompositions more “romantic” than the master

00

250r=5 cm

r=10 cm

5 10R

BF (min−1)

R

M

M

r = 10 or 5

m

Figure 1. The beat frequency, BF, of ametronome as a function of the distance R of theheavy weight (M=40 g) from the pivot axis attwo fixed positions of the movable weight: r= 5cm and r= 10 cm. As is evident, the beatfrequency will be extremely sensitive to a smallchange in the position of the heavy weight—inparticular for small values of r.

intended or else given up as being technically toodifficult?

The beat rate was indicated by gradations on theoscillating beam. Beethoven’s eyesight was not thebest. He could have read the wrong numbers. Onthe other hand, he often had help from his nephewKarl, who may not have had vision problems. Thenthere is the connection between the beat rate andthe note length: a beat rate of “60” could referto the length of a whole note, a half note, or aquarter note, and the tempo of the music changesaccordingly. A beat rate of 60 for a whole notebecomes 120 for a half note. Accidentally removingthe stem from the sign of the half note wouldseemingly double the tempo. Could copy errorshave been the culprit?

We should also consider psychological factors.Different humans may have slightly different“internal clocks”. A tempo regarded as fast for oneperson may be less so for another. We supposethat internal clocks have a tendency to run slowerwith age in most humans, one notable exceptionbeing Toscanini’s, whose clock seemed famouslyto never lose time.

But one must also consider reports thatBeethoven’s metronome on occasion was out of

October 2013 Notices of the AMS 1149

BFapparent

BFint

correct metronomeR = 5

damaged metronomeR = 7

BF(min−1)

200

900 5 10

M = 40gm = 8g

r1r2r

Figure 2. The beat frequency, BF, of ametronome as a function of the distance r fromthe pivot axis of the movable weight (m = 8 g) .

The heavy weight (M= 40 g) is assumed to be atthe distance R= 5 cm from the pivot axis for the

correct, factory-calibrated metronome and at thedistance R= 7 cm for the damaged metronome.

The damaged metronome will beat slower thanindicated by the factory-engraved markings for acalibrated and correct metronome when r is less

than 8 cm, while it will go faster than theengraved markings at r larger than 8 cm. Thuswhen r < 8 cm, BFdamaged < BFcorrect; when r = 8

cm, BFdamaged = BFcorrect; when r > 8 cm,BFdamaged > BFcorrect.

order. In an April 1819 letter to his friend andcopyist Ferdinand Ries at the Fitzwilliam Museumin Cambridge, Beethoven states that he cannot yetsend Ries the tempi for his sonata Op. 106 becausehis metronome is broken [11]. Beethoven was veryirritated and upset, perhaps from symptoms oflead poisoning, a condition he may have hadaccording to studies at the Argonne NationalLaboratory [12]. Could Beethoven on occasion havedropped the metronome on the floor or could hehave used more than one metronome in his lastten years?

Peter Stadlen is reported to have foundBeethoven’s metronome after a long search. Thisparticular specimen was among the propertyauctioned off by his surviving nephew Karl afterhis uncle’s death. To Stadlen’s disappointment theheavy weight was gone, perhaps an indication thatthe early metronomes from Mälzel’s factory didnot have rugged, shock-proof construction.

Dynamical Properties of the MetronomeFinally, we take a closer look at the dynamicalproperties of the mechanical double pendulummetronome and, in particular, determine if itsperformance is sensitive to the position of thesupposedly fixed heavy weight with respect to thepivot. (See the text below on the derivation of theequations of motion.)

The final result for the oscillation frequency Ωof a Winkel-type double pendulum is as follows:

(1) Ω = [ g(MR−mr)(MR2 +mr2)

]1/2.

Here M is the mass of the heavier fixed weightbeneath the pivot along with its distance R tothe pivot axis. Correspondingly, m is the mass ofthe movable weight at distance r above the pivotpoint. The lighter weight and its distance are whatmusicians can see, while the heavier weight isusually hidden in the base of the cabinet. Also, g isthe acceleration due to gravity (9.81 m/sec2). ButΩ is not the same as the beat frequency in “clicksper minute” of the metronome (BF). As can be seenin the derivation, the beat frequency BF is relatedto Ω by the ratio(2) BF = 60Ω/π.The equations are valid under the assumptionthat there is no friction or drag, the masses arepoint masses, and the amplitude of the oscillationsaround the vertical position is small. The lastassumption is not severe as the beat frequencywould not deviate much from the above expressionfor relatively large amplitudes; however, the correctequations would be far more complex and nonlinear.Please note that BF is the limit when r goes tozero, and it is independent of the mass of thetwo weights as determined by R, or rather 1/

√R,

which is a consequence of m being located on thepivot axis with the double pendulum beating like asimple pendulum.

Equation (1) gives us an opportunity to investi-gate the sensitivity of the metronome frequencyto a change in parameters. For any particularmetronome we can assume that both M andm arefixed at the factory by the manufacturer. The valueof R for the heavy weight is also likely to be set atthe factory in order to make the metronome beatat the correct frequency when the moving weight isput at the corresponding grading on the oscillatingbeam. But what if an error had been made? Perhapsthe heavy weight was attached to the beam in sucha way that it actually could slide—move—from itsoriginal position. Suppose the metronome acciden-tally fell to the floor or was otherwise damaged.We could investigate what the consequences wouldbe. But first, just for curiosity’s sake, let us look athow the beat frequency of a metronome in clicks

1150 Notices of the AMS Volume 60, Number 9

per minute, BF, will depend on the position of theheavy weight in relationship to the lighter weightat a fixed distance. In other words, suppose onlythe location of the heavy weight M is changed.

We now turn to numerical results. Althoughwe have searched extensively, we have not foundthe actual values of the parameters M , m, and Rthat Beethoven used to obtain his tempo markings.Over a limited parametric space, however, theresults are not sensitive to the values picked. Forthe sake of argument, let’s assume M = 40 gramsat distance R = 5 cm for the heavier weight andm = 8 at distance r = 10 for the upper weight.(See Figure 1.) We note that the beat frequencyBF of a metronome is extremely sensitive to theposition of the heavy weight if the distance R isless than about 3–4 cm and BF goes to zero atR = 1 cm and r = 5 cm. At the latter distances thetwo weights exactly balance each other with thecross product being (40)(1) = (8)(5). Under theseconditions the metronome beam would rotatefreely like a balanced propeller if there were nophysical constraints. For r =10 this condition ismet when R = 2; i.e., (40)(2) = (8)(10). As illustratedin Figure 1, the heavy weight should not be placedtoo close to the pivot as it is important to minimizethe effects of small changes in its position. But itis also evident that, for R values larger than thoseat the maxima in the arcs of the curves, shiftsin the position of the heavy weight toward largerdistances from the pivot will result in BF shifts tosmaller values. This effect is particularly strikingwhen the movable weight is close to the pivot, i.e.,in the region of fast tempi of the metronome!

Finally, let us look at what would happen if themetronome were somehow damaged by a shiftin the heavy weight from its calibrated factoryposition. We consider two cases. First we willassume that the heavy weight is moved such thatit is a longer distance (RBeethoven) from the pivot, asituation that could occur if the metronome fell tothe floor in an upright position. (We hope that HerrLudwig, as he rests in heaven, will forgive us for thenotation (RBeethoven).) If we take (RBeethoven) to be7 cm and assume that the original factory positionwas 5 cm, the result is shown in Figure 2. For fasttempi (BF axis), the damaged metronome will goslower than indicated by the factory calibration.At a medium value of r we reach a point where thefactory gradations happen to be correct, but forslow tempi (see R) the damaged metronome alwayswill go faster than predicted by calibrations.

In the second case let us assume the heavyweight is moved to a position closer to the pivotthan its original factory location, perhaps becausethe metronome had accidentally fallen to the floorwith its top down. In Figure 3 the results aredisplayed for a case in which the factory position

050

110

138

M = 40gm = 8g

250

5 10 15

r2 r1 r

BFobserved

BFintended

BFpublished

correct metronome

damaged metronome

BF (min−1) Rg = 6cm; Rv = 4cmx max = 17 cm

y min = 50; y max = 250

[cm]

ρ = 138

Figure 3. The beat frequency, BF, of ametronome as a function of the distance r fromthe pivot axis of the movable weight (m = 8 g).The heavy weight (M(M(M = 40 g) is assumed to be atthe distance, R= 6 cm from the pivot axis for thecorrect, factory-calibrated metronome and at thedistance R= 4 cm for the damaged metronome.Beethoven puts the movable weight on hismetronome to correspond to the marking BF=110 ( i.e., BFint in Figure 3) but, somewhatpuzzled perhaps, finds that the visibly observedBF seems far too slow, around seventy to eightyor so, i.e., the BF of the damaged metronomecurve at the setting r1r1r1 (the point indicated byBFobs.]

of the heavy weight is 6 cm from the pivot (R = 6cm) and the distance after an accident is R = 4 cm.We note by looking at Figure 1 that, in the region ofsmall values of R, even a small change in positionwill have a major effect on the metronome’s beatfrequency. Other sets of parameters produce evengreater changes.

Could these results explain Beethoven’s “ab-surdly fast” metronome markings? We should notethat, if the same model is applied in the region ofslow tempi (see Figure 2), the printed markingswould result in somewhat slower beat frequencieson the damaged metronome than intended by themaster.

Let’s envision the following hypothetical scenario(cf. Figures 1–3). Unknown to him, the metronomeBeethoven is working with is damaged in the sensethat the heavy weight hidden by the wooden casehas been displaced. Assume Beethoven puts themovable weight on his metronome to correspondto the marking of approximately BF =110 (i.e.,

October 2013 Notices of the AMS 1151

BFintended in Figure 3). Somewhat puzzled perhaps,he finds the visibly observed BF seems far tooslow, around 70 to 80, i.e., the BF of the damagedmetronome curve at the setting r1 (the pointindicated as BFobserved). The markings on themetronome beam with the light movable weightthat he can clearly see do not correspond to hisdesired BF. Beethoven, dissatisfied with the slowmovement of the visible metronome beam, thenmoves the weight until he is satisfied with themuch higher BF.

Today we know very little about who actuallyhelped the master with the practical details of pro-ceeding from his raw, hardly legible, handwrittensheets of music, but we are very sure that a BF = 138was published (p=138). (See Figure 3.) A latter daypianist intending to perform the Hammerklaviersonata looks at the printed sheet of music markeda “half-note = 138,” sets a correctly calibratedmetronome, and mutters, “incredibly fast”—veryfast indeed! Our scenario provides an explanationfor Beethoven’s “fast tempi problem”, if indeedthere is one.

How could Beethoven not note the occasionalodd behavior of his metronome? A thoroughaccount by Peter Stadlen gives the impressionthat the master was not entirely comfortable withthe new device, most especially in the process ofconverting from beat frequencies to actual tempimarkings for half-notes, quarter-notes, etc. [13].Obviously, it would be very helpful if we knew moreabout the actual design of his metronome(s). Wesuggest that one or more of the devices could havebeen damaged, perhaps accidentally during one ofhis well-known violent temper tantrums. Whateverthe case, our mathematical analysis shows thata damaged double pendulum metronome couldindeed yield tempi consistent with Beethoven’smarkings.

Arguably this is a bold hypothesis. Perhapssomeone else was involved in the procedure—Beethoven’s eyesight was not always the best.

Derivation: Equations of Motion for a DoublePendulum for the Type Used in MechanicalMetronomes.Consider the following model of the doublependulum:

We will assume that the beam oscillates aroundthe pivot without any friction and also that themass of the beam can be ignored in comparisonto that of the two weights. The total energy E ofthe oscillating pendulum is a sum of its potentialenergy V and its kinetic energy T .

The kinetic energy may be written

(3) T = 12m(vm)2 +

12M(vM)2

where vm and vM are the velocity of the light andheavy masses, respectively, in the direction of theirmotions. Now

vm =d(Sm)dt

where (sm) = r sinΘ,vM =

d(SM)dt

where (sM) = R sinΘ.For small values of Θ we have sin Θ = Θ (in radians).Thus

sm = rΘ and sM = RΘ.The velocities v are then

vm = rdΘdt= rΘ̇ and (vm)2 = r2Θ̇2,

vM = RdΘdt= RΘ̇ and (vM)2 = R2Θ̇2.

The kinetic energy of the double pendulum is thus

(4) T = 12

[mr2Θ̇2 +MR2Θ̇2] .

Now let us consider the potential energy V withrespect to the axis of the pivot:

(5) V = gmr cosΘ − gMR cosΘ,where g is the acceleration due to gravity. Againassuming small angles of oscillation when sin Θ= Θ and since sin2Θ + cos2Θ = 1, we can replacecos Θ with (1−Θ2) 12 . The potential energy V thenbecomes

(6) V = g[mr−MR](1−Θ2) 12 .But, if the double pendulum moves without friction,then the total energy E is constant and the timederivative will be zero:

dEdt= 0.

Let us for the sake of clarity take the timederivatives of T and V separately:

dTdt= 1

2

[mr22]Θ̇Θ̈ +MR22Θ̇Θ̈](7)

=[mr2 +MR2

] Θ̇Θ̈where Θ̈ = d2Θdt2 .

Now consider the time derivative of the potentialenergy for small oscillations:

dVdt= ddt[g[mr−MR](1−Θ2) 12 ].

Note that

d√(1−Θ2dt

= 12

(−2ΘΘ̇√(1−Θ2)

)≈ −ΘΘ̇ for small Θ.

1152 Notices of the AMS Volume 60, Number 9

Continuing,

dVdt= g(MR−mr)ΘΘ̇,

and sincedEdt= ddt[T+V] = 0,

(8) (mr2 +MR2)Θ̇Θ̈ + g(MR−mr)ΘΘ̇ = 0.We can eliminate Θ̇ since Θ̇ = 0 does not giveanything, and we are left with

(9) Θ̈ + g[ (MR−mr)(mr2 +MR2)

]Θ = 0.This is a differential equation of the type

d2xdt2

+ a2x = 0

with the solution

x(t) = (x0) cos(ax+ a),and if we introduce

Ω = [g(MR−mr)MR2 +mr2)

] 12

,

the solution to equation (9) is

(Θ(t) = (Θ0) cos(Ωt +Ω).Let us check to see if this expression yields thesame for a simple pendulum if we put m = 0. Wehave

Ω = [g ( MRMR2

)] 12

=[gR

] 12

,

Θ(t) = cos[(g/R)t], and thus the oscillation fre-quency is Ω = (g/R), or the correct equation forsmall amplitude oscillations. We note that Ω is infact the oscillation also for our double pendulum!

How do we relate Ω to the beat frequency of themetronome? The period of any type of pendulumis the time it takes to swing from one startingposition and back to that position again. But thiscorresponds to two “clicks” of the metronome—itclicks at every turning point of the oscillatingbeam! Therefore, to go from pendulum period(P = 2π/Ω) to beat frequency “BF” in “clicks perminute” we have

BF = 60Ω/π.References[1] P. Stadlen, Music & Letters 48, no. 4 (1967), pp. 330–

349.[2] L. Talbot, J. Sound and Vibration 17, no. 3 (1971), pp.

323–329.[3] P. A. Scholes, chapter on notation, The Oxford Com-

panion to Music, 10th ed., Oxford University Press,1975, pp. 688–689.

[4] T. Y. Levin, The Musical Quarterly 77, no. 1 (1993),pp. 81–89.

[5] http://curvebank.calstatela.edu/birthdayindex/apr/apr14/apr14huygens/apr14huygens.htm.

[6] R. Kolisch, The Musical Quarterly 77, no. 1 (1993),pp. 90–131.

[7] http://www.randi.org/jr/02-03-2000.html.[8] J. Suchet, http://www.madaboutbeethoven.com/.[9] Beethoven’s Letters 1790–1826 in translation by Lady

Wallace.[10] http://www.benjaminzander.com/recordings/

bpo/beet9.asp.[11] E. Anderson, Music & Letters 34, no. 3 (1953), pp.

212–223.[12] W. Clairborne, Washington Post, October 18, 2000,

p. A03.[13] P. Stadlen, Beethoven and the metronome, Music &

Letters 48, no. 4 (1967), pp. 330–349.[14] New Grove Dictionary of Music and Musicians, 2nd ed.,

16, Macmillan, 2001, pp. 531–537.[15] The entire sonata is online: http://mill1.

sjlibrary.org:83/record=b1011883 S0.

About the Cover

Baltimore, MD, is the location of the 2014 Joint Mathematics Meetings. Pictured on the cover is a nighttime view of the Baltimore Inner Harbor.

One item of interest in Baltimore for math-ematicians is that the Walters Museum was the site of work on the Archimedes Codex. In response to a query of ours, however, Nancy Zinn, deputy director at the Museum, reports that the codex is no longer there. She says further:

“Since the completion of our analysis and conservation of the manuscript, the codex has been returned to its owner, and is no longer on deposit at the museum. However … the owner has agreed to our request for the palimpsest to be on view at the Huntington Library in San Marino, CA, from March 15 to June 8, 2014. This will be the third, and final, venue for the exhibition ‘Lost and Found: The Secrets of Archimedes’. It is unlikely that the codex will ever be displayed publicly again.”

A complete report about the palimpsest can be found at

http://archimedespalimpsest.org/

—Bill CasselmanGraphics Editor

(notices-covers@ams.org)

October 2013 Notices of the AMS 1153

http://www.madaboutbeethoven.com/http://www.benjaminzander.com/recordings/bpo/beet9.asphttp://www.benjaminzander.com/recordings/bpo/beet9.asphttp://archimedespalimpsest.org/http://curvebank.calstatela.edu/http://curvebank.calstatela.edu/http://www.randi.org