Mar 20, 2015 - references to the sound-testing of coins(II); in the third part, I use a thin ..... Napoleon III with naked head (1860) Pt + Au plated 6.45 g 6,320 Hz ...
The Music of Gold Can gold counterfeited coins be detected by ear? Arnaud Manas March 20, 2015 Abstract — In this paper I investigate whether it is true and to what extent counterfeit coins can be detected by their sound frequency. I describe the different types of counterfeit coins encountered and their respective characteristics. I then use the Kirchoff thin plate theory to models a coin, and confirm the validity of the theory by listening to the tone of genuine and counterfeit coins. Keywords: Kirchoff model, Gold counterfeit coins, eigenfrequencies, vibrating plates
Introduction “If I were writing The Counterfeiters I should begin by showing the counterfeit coin - the little ten-franc piece you were speaking of just now.’ So saying, he pulled out of his pocket a small coin, which he flung on to the table. ’Just hear how true it rings. Almost the same sound as the real one. One would swear it was gold. I was taken in by it this morning, just as the grocer who passed it on to me had been taken in himself he told me. It isn’t quite the same weight, I think; but it has the brightness and the sound of a real piece; it is coated with gold, so that, all the same, it is worth a little more than two sous ; but it’s made of crystal. It’ll wear transparent. No; don’t rub it; you’ll spoil it. One can almost see through it, as it is.”, Andr´e Gide, The Counterfeiters, Penguin, 1966, p.172 Gide describes a counterfeit coin made out of gold-plated crystal which makes the same ringing sound as a genuine gold coin. From a practical point of view, of course, such a coin is complete nonsense: crystal density is around 3 while lead is 11 and gold 19; moreover, a coin like this would be impossible to mint and would instead have to be moulded or hand-engraved. Gold-plated crystal is used here as a literary metaphor for false values, transparency and perversity. However, Gide’s description does raise the question of whether it is true or just a myth that genuine coins can be detected by the sound they make. A number of websites advise making a coin resonate to test whether it is fake. These ”ping”
1
or ”ring” tests are even available as smartphone apps, such as Bullion Test 1 and Cointrust 2 . In this paper, I attempt to test whether or not it is possible to detect counterfeit gold coins by their sound. It is divided into five parts: the first part describes the types of counterfeit coins encountered and their methods of production (I); the second part looks at traditional methods of detection and presents different references to the sound-testing of coins(II); in the third part, I use a thin vibrating circular plate to simulate a gold coin within the Kirchhoff theory (III); in the fourth part, I perform experiments on both genuine and counterfeit coins (IV); lastly, I recognize the musical notes produced by platinum and gold coins (V).
1
Counterfeiting
The main goal of the counterfeiter is obviously to make a profit while avoiding punishment, although it is true that some counterfeits have been produced for political motives3 . As Reiff [23] stresses, “The forgers’ principles were to avoid immediate discovery of the falsification as well as the tracking of its origin and to realise the highest possible profit”. The counterfeiter’s profit is the difference between the face value of the genuine coin and the production cost of the fake coin. For coins containing a metal with an intrinsic value, such as gold or silver coins, the face value corresponds to the cost of the metal plus the minting fees4 . Therefore the counterfeiter’s profit derives mainly from the difference between the cost of the genuine coin metal and the cost of the metal used for the counterfeit coin, as the counterfeiter’s own production cost are more or less equivalent to the brassage and his protection costs can be assimilated to seignoriage. Hence the counterfeiter has an incentive to use the cheapest possible metal or combination of metals while maintaining the appearance of a genuine coin. For gold coins, he has two options: he can produce either homogeneous or composite coins. In the first case, he will use a low gold-content alloy5 containing 25% to 1 https://play.google.com/store/apps/details?id=se.brolinembedded.bulliontestfree “This app offers a cheap and easy to use method to help detect fake bullion coins, by examining their resonance frequencies. The resonance frequencies of an object are determined by the objects shape and material. If the shape and weight of a silver coin is correct, but the material is not actually silver, then a resonance test could detect this. This is also known as a ”ring test” or ”ping test”. Unlike the manual ”ring test”, this app does not require that the user knows exactly how each coin should sound. The app will also detect resonance frequencies above the range of human hearing.” 2 http://www.cointrust.com/ 3 During WWII the Nazis tried to destabilise the United Kingdom by flooding it with forged Bank of England notes (Operation Bernhard) 4 The minting fees “consisted of two distinct elements: brassage, for the mint-master, and seigniorage, for the prince (or republican government).[...] Brassage was an economically necessary fee for the simple reason that it literally ‘costs money to make money’. That fee thus compensated the mint-master for his own production costs: above all, the cost of the copper alloy; the capital costs of maintaining the mint, in the form of his dye tools (hammers), the furnaces, forges, melting pots, shears; the administrative costs of managing the mint including light, heat, and rent; the labour costs of producing the actual coins and running the mint; and finally the cost of licence fees paid to the prince and other fees paid to official coin assayers and mint inspectors. Such costs were normally relatively modest.” [18] 5 Some counterfeit coins with a fineness of 750 have been found. This points more to a fraud within the mint rather than to counterfeiting. During medieval times, royal mints secretly
2
66% gold instead of the 90-92% standard alloy for genuine coins. The design is reproduced using one of two methods: it is either cast or struck. In the second case, the counterfeiter produces a “sandwich” coin with a core made of cheap “base” metal. The core is then coated with gold or gold alloy, or inserted into a hollowed out genuine coin6 . The gilding techniques have been described by Oddy [19] (see also La Niece [12]). Basically, there are four main techniques: • Foil/leaf gilding: the core is covered with a gold foil or a gold leaf7 . Several attachment techniques have been described (diffusion bonding, cold mercury “soldering”) • Fire gilding (“ormolu8 ”): a gold mercury amalgam is applied to the base metal core and the mercury is evaporated by heating. • Electrochemical gilding: a copper or silver core is immersed in a gold salt solution. • Electrolytic gilding (galvanization) The counterfeiter chooses the base metal on the basis of its properties and cost. It needs to meet a number of conditions, notably the weight of the forged coin must be realistic. Until the discovery of heavy metals like platinum9 and tungsten10 gold was the densest of all known metals. This property was a strong security feature as the coin could be weighed or its specific gravity determined using Galileo’s hydrostatic balance, based upon Archimedes’ principle. Therefore the counterfeiter had to use heavier metals like lead or silver or secretly increase the coin’s thickness. Several assayer’s manuals and authors ([6], [7], [4]) from the 19th century describe the most common types of counterfeit encountered.
2
Detection of counterfeits
The seller who unwittingly receives a false coin is a victim of information asymmetry11 as he does not know whether the buyer is honest or dishonest. Nevertheless,depending on the prevalence of counterfeits at that time, he can adjust his behavior and perform a number of cost effective tests based on his own knowledge, the instruments at his disposal and the amount of time he has available. Eckfeldt suggests three levels of testing12 . lowered the gold content of the official alloy used in their coins (“debasement”), see Mongez [17] and [6] 6 Eckfeldt[6] notes that these counterfeits are made by “Sawing out the interior of a good coin and leaving two very thin, but genuine outside disks, soldered upon a base body” 7 Gold leaf is so thin (0.1 µm) that it cannot support its own weight unlike gold foil (over 10 µm). 8 See Delamare 9 Platinum was discovered in 1736 and has been used since the 19th century. Between 1828 and 1845, the Russian imperial government minted platinum coins. Platinum was worth one third the value of gold (by weight) see [6], p.110 10 Tungsten was discovered in 1781 but was not widely used before 1915, see [14] 11 This is Akerlof’s classical Lemons problem see [1] 12 “The principal object of this chapter is to enable persons to discriminate between true and false coins. The need of such information is felt in various circumstances in life, but the most important are these three : 1. When a doubtful piece of money is offered in some such place
3
Coin 48 livres 48 livres 48 livres 24 livres 24 livres 40 francs 40 francs 20 francs Eagle 1801 Eagle 1810 Half eagle Doubloon Doubloon Escudo Escudo 40 francs
Method Gold and copper Gilded Copper Platinum covered with gold Gold, silver and copper Gilded Copper Gold and copper Silver covered with gold Gilded silver Silver with gold surface Silver with gold surface Gold, silver and copper Gilded silver Gold, silver and copper Gold, silver and copper Gold, silver and copper “outsides of actual coin”
Characteristics Au .830 15.29 g instead of 16.25g Au .21, 7.22 g instead of 16.25g 15.33g instead of 16.25g Au .525 6.95g instead of 8.12g Au .20 5.24 g instead of 8.12g Au .450 10.21 g instead of 12.90 g Au .20 4.16g instead of 6.45g Dull sound, 179 grains Sonorous sound, 204 grains Au .440, sharp sound, 115 grains Good sound, 342 grains Good sound, 368 grains Au .510, good sound, 46 grains Au .257, good sound, 36 grains Dull sound, 171 grains
Table 1: Common counterfeits • Level 1 - sensory tests (sight, smell, hearing and touch) • Level 2 - mechanical tests (paring with a knife, filing the edge, touchstone, scales, hammer and chisel, specific gravity) • Level 3 - chemical tests (assays by cupellation). This three-level classification is still in use. For example the ECB recommends the following methods13 : 1) Feel-Look-Tilt (unaided test); 2) testing of additional features with specific equipment : UV, microprint (needs a magnifying glass), infrared; 3) secret features with laboratory equipment (these tests are not documented, for obvious reasons). Eckfeldt classifies the sound test as a level 1 test: “The Sound is a more certain criterion. To apply it, the coin should be poised on the tip of the finger, and its edge lightly struck by another coin. (Dropping it upon a stone is not a good method.) The tone of standard gold or silver is sonorous, full, and agreeable. That of baser metal is sometimes shrill and short, sometimes flat and leaden. In this experiment, a genuine coin should always be used for a comparison. After all, this proof is not greatly to be relied upon. It would be more so, if the practice were universal which is said to be pursued at the London mint, of trying every coin by its ring, and as the market, or at a counter, and consequently very little time is afforded to decide upon taking it ; in which case, the simplest tests only can be resorted to. 2. Where the piece in question is of a large denomination, and much interest is felt in ascertaining its true character ; in such case, more time and pains can be taken. 3. When a person is under suspicion or arrest for the willful forging or uttering of counterfeit money ; or when evidence is to be given on a trial for that crime, before a court. In such case, the common tests may not be sufficient, and resort is to be had to the severest scrutiny”.[6] 13 https://www.ecb.europa.eu/euro/banknotes/security/html/index.en.html
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Ref. [4] 1 [4] 2 [4] 3 [4] 4 [4] 5 [4] 14 [4] 15 [4] 16 [6] 1 [6]2 [6] 3 [6] 4 [6] 5 [6] 8 [6] 9 [6] 10
allowing none of uncertain sound to go into the circulation. As it is, pieces occasionally escape from other mints, and our own too, with a slight and often imperceptible flaw, which obstructs the vibration, and deadens the sound. But even a good sonorous piece may be so maltreated, after it leaves the mint, as to become unmusical.” (Eckfeldt, p. 173) He classifies gold counterfeits as “Dull, Sonorous, Sharp or Good”. Mongez claims that this method was used by the Greeks and the Romans, and quotes Epictetus (“one drops the coin several times and listens to its sound”). Chaudet notes that unfortunately sound is seldom used as a criterion in France; in Great-Britain, however, its use is widespread, and all gold coins are systematically rung. Nevertheless, he acknowledges that some genuine coins can produce a dull sound because they have become oxidised or flawed. Modern experts14 acknowledge that coins made out of gold or silver generally emit when they are rung a longer-lasting, higher-pitched sound “like a bell” [26] while coins made out of base metal emit a shorter-lasting, lower-pitched sound. Nevertheless, he warns that genuine silver coins may not ring true due to crystallization of the silver resulting from intergranular corrosion, reticulate corrosion, granularisation, or embrittlement15 . Apparently, these problems do not occur for gold alloys over 14 ct in normal conditions.
3
Kirchhoff model
All objects vibrate at their own specific frequencies. From a physical point of view, coins are thin plates, with a small thickness in comparison with their width16 . It is therefore possible to use Kirchhoff’s classical thin plate model, developed in the mid-19th century. This model describes the transverse vibrations (T-waves) producing the radiating sound ([25] p.71); the longitudinal waves play no role in the ringing. The modes are determined by the eigenvalues which are the roots of the frequency equation. This equation depends on shape and boundary conditions. For a circular plate, it involves Bessel functions and must be solved numerically17 . The standing waves associated with the mode frequencies form Chladni figures along diameters and circles (see figure 1). The fundamental frequency (i.e. lowest) for a disc is f2,0 where two perpendicular diameters remain still. In the case of a free standing disc the eigenvalues depend on the Poisson’s ratio of the alloy used, whereas in the case of a clamped disc they are independent of the Poisson’s ratio. The relation between the vibrating frequency and the eigenvalue is18 14 http://www.sid.in-berlin.de/nedkelly-world/counterfeitingcoins1.htm 15 On
embrittlement of ancient silver, see [31] thickness to width ratio is under .05 17 See [15] for equations and references. It should be noted that Leissa’s equation (2.14) has a typo: λ3 In0 (λ) should read λ3 Jn0 (λ). 18 Where E is the Young’s modulus, h the disc thickness, a its diameter, ν the Poisson’s ratio and ρ the mass density. 16 The
5
Figure 1: First modes (respectively f2,0 , f0,1 and f3,0 )
f=
h 2 2πa | {z }
shape factor (qs
s
E λ2 12ρ(1 − ν 2 ) |{z} {z } eigenvalue | )
(1)
metal factor (qm )
Hence the possible frequencies for a coin - considered as circular plate depends on three factors: a shape factor depending only on the geometry of the coin and its design; a metal factor depending only on the physical properties of the metal; and the eigenvalues. Though the influence of the design on the frequencies can be neglected in a first approximation, it has a dual effect on resonance. First, it modifies the overall frequency distribution: the frequencies for a given coin will be slightly different from a the ones emitted by the unstruck coin (blank). Second, it breaks the axial symmetry of the disc and splits the frequencies involving diameters (fn,c where n ≥ 1) because of the different possible orientations19 . The gold 20 franc coin is known as the Napoleon because it was first minted in 1803 when Napoleon Bonaparte, who was First Consul at that time, reformed the French monetary system (Law of Germinal 1803). He decided to produce new 20 franc coins made out of gold with a fineness of .900 . One kilo of this gold produced 155 of the new Napoleons. The coins were exported all over Europe during the Napoleonic wars. They were used in Italy, Belgium and Switzerland. In 1865, a treaty was signed between France, Italy, Switzerland and Belgium creating the Latin Monetary Union and allowing standardised national coins to be used in all the participating countries. Eventually more than 20 countries joined the union and over 100 different types of coins were in circulation20 prior to World War I. The shape factor is easy to estimate for the Napoleon: qs = 1.44 (height= 1 mm, diameter = 21mm). For the metal factor several aspects need to be taken into account. First, the alloy changed over time.Initially, the French Mint used a gold silver alloy (Au .900 - Ag.100), it then switched to a gold-copper alloy (Au .900 - Cu .100) in the first quarter of the century (see [17]) . The striking technology also evolved: the first coins were struck using a screw press, but this was subsequently replaced by the steam powered knuckle press21 . In 19 On this subject, a parallel can be drawn with musical mugs where the handle breaks the axial symmetry, see [10] 20 see http://www.unionlatine.com 21 In France, the switch to the Thonnelier press took place in 1845 (see [27] p.64 and [13]p.145)
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Figure 2: First gold 20 franc coin : Year XI (1803), Bonaparte First Consul addition, the mints used non documented metallurgic treatments to alter the properties of their coins. Furthermore, the physical properties of monetary alloys are not readily available. The density (ρ) is comprised between 17.9 for Au-Ag alloy and 17.2 for Au-Cu (see [6] and [11]). The Poisson’s ratio (ν) should be between .35 and .45 and Young’s modulus (E) between 80 and 120. This implies a metal factor of around 1,100 within a ±5% range. Table 2 gives the standard physical data22 for metals used in counterfeits; the metal factor qm is directly computed from these data according to equation (1). Table 3 gives the theoretical eigenvalues and frequencies computed23 according to Kirchhoff model for the three first modes. The theoretical frequencies for the Napoleon are around 5,500 Hz for f2,0 and 10,090 Hz for f0,1 .
Platinum (Pt) Gold (Au) Tungsten (W) Lead (Pb) Silver (Ag) Copper (Cu) Tin (Sn) 20 franc
Density ρ (Kg/dm3 ) 21.09 19.30 19.25 11.34 10.49 8.92 7.31 17.2/17.9
Young’s Modulus E (GPa) 168 78 411 16 83 130 50 80-120
Poisson’s ratio ν .38 .44 .28 .44 .37 .34 .36 .35-.45
Metal factor qm 881 646 1,389 382 874 1,172 809 1, 100 ± 50
Table 2: Metal properties 22 See http://www.webelements.com/ which relies on A.M. James and M.P. Lord in Macmillan’s Chemical and Physical Data, Macmillan, London, UK, 1992 and G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993 23 The frequency equations has been solved using Excel Solver and the built-in BESSELI and BESSELJ functions.
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Metal
qs qm
λ2,0
λ0,1
λ3,0
Pt Au W Pb Ag Cu Sn 20 franc
1,272 933 2,006 551 1,262 1,692 1,168 1,100
2.25 2.21 2.33 2.21 2.26 2.29 2.27 2.24
3.02 3.05 2.98 3.05 3.02 3.01 3.02 3.03
3.38 3.83 3.54 3.83 3.46 3.50 3.47 3.43
f2,0 (Hz) 6,466 4,548 10,871 2,687 6,467 8,840 6,025 5,500
f0,1 ( Hz) 11,651 8,661 17,986 5,118 11,538 15,367 10,662 10,090
f3,0 (Hz) 14,527 13,685 25,193 8,086 15,088 20,687 14,050 12,880
Table 3: Theoretical frequencies for a disc (qs = 1.44) of metal
4 4.1
Testing the coins Methodology
The following methodology was used to test the coins. First the coins were struck with a hard object to transfer kinetic energy, in turn activating one or several vibration modes; this energy was then dissipated through the emission of sound waves with the corresponding frequencies and with an exponential decay. The literature suggests three different ways of doing this: according to Anderson [2] who uses Eckfeldt’s method [6], the coin should be laid horizontally on the pad of the finger and should be hit with another coin in a downwards movement; another method24 is to lay it on one finger and ping it with a plastic pen; lastly the coin can be dropped onto a thick hard surface25 . Two other methods were considered: one was to glue or solder the center of the coin to a rubber band and hit it with several objects26 . This method proved impractical because it would have damaged fake gilded coins. The other method was to hang the coin from a thread (see figure 3) and to hit it with a heavy metal rod (in fact an old brass key27 weighing 28 g). This method proved very effective (see infra). The sound was recorded with Audacity a free open source digital audio software that allows you to record and compute the Fourier transform of a sound recording with an FFT algorithm. The sampling rate was set at 192 KHz allowing a 96 KHz maximum signal frequency according to the NyquistShannon sampling theorem, well above the human audio frequency range (20Hz20KHz). The noise of the shock was removed to restrict the analysis to the actual resonance frequencies. In practical terms, the first 2-5 first ms of each recording were removed. 24 see http://metauxprecieux.blogspot.fr/2009/02/fausse-monnaie-une-methode-simplede.htm under the pseudonym Argentum Aurum. 25 This method is suggested by [29] and was used by the Banque de France until the 1970s: a special thick block (4cm × 14cm × 14cm) of stone or ceramic was used to make the coins ring: this block was known as a “tas” (which literally translates as ”a heap”) and even appeared in some dictionaries as a metal device to test the sound of coins. Eckfeldt [6] does not consider this method to be efficient. 26 This method would be equivalent to a free standing coin (see footnote supra). 27 All experiments have been conducted with standard household equipment: another identical coin, a plastic chopstick, a mantelpiece and a key.
8
Figure 3: Coin suspended with a thread
4.2
Genuine coins
Four 20 franc coins were used: a Swiss 20 franc ( bearing the head of Helvetia and dated 1890), an Austrian one (bearing the emperor Franz Josef and dated 1882), and two French coins (one with First Consul Bonaparte from 1805 and one with King Louis-Philippe from 1839). The Swiss coin, the most recently minted of the four coins (see table 4.2) , produced the sharpest results and was therefore used as a benchmark. The four methods (see supra) were tested and themethod where the coin was suspended from a thread produced the best results. The plastic rod was too light and insufficient energy was transferred to the coin to be propagated in the form of sound waves; therefore this method was discarded. The method of placing the coin on the finger and pinging it with another coin or a metal rod gave mixed results. The coin’s natural frequencies appeared in the spectrum but several other lower frequencies were also present. The origin of these frequencies seems to lie in the fact that the finger acts as a small elastic cushion, as it does not completely cover the coin, the coin’s behavior is a mix of that of a free standing coin and that of a coin resting on a Winkler. Hence the appearance of two sets of frequencies. The description of the full spectrum would imply solving the equation of a partial Winkler foundation (see [22]). The method of dropping a coin onto a block produces a different set of frequencies caused by the edge of the coin spinning on the block. The fourth method (coin hanging by a thread) and hit with a heavy metal rod produces good results. Furthermore, it seems that it is possible to activate specific modes by hitting the coin in given places. Method Coin on finger pinged with a coin
Coin on finger pinged with a plastic rod Coin dropped onto a massive block
Coin on thread pinged with metal rod
Frequency 2,880 Hz/ 4,240 Hz 5,060 Hz/ 5,326 Hz 9,316 Hz Noisy signal 1,854 Hz /3,539 Hz 5343 Hz / 8,727 Hz 9,284 Hz / 10,552 Hz 5,390 Hz / 9,310 Hz
Table 4: Method used with Helvetia 1890 It should be noted that the natural frequencies for the Swiss coin (see table 4.2) correspond to the physical parameters ν = 0.33 and E = 92which are 9
Hz 0
1 000
2 000
3 000
4 000
5 000
6 000
7 000
8 000
9 000
10 000
11 000
12 000
0
-20
-40
dB
-60
-80
-100
-120
-140
-160
Figure 4: Frequency spectra (db / Hz) for the Helvetia 1890 suspended from a thread plausible28 . As noted earlier the fact that the coin does not have a perfect axis of symmetry due to its design splits the fundamental mode. These modes can be activated by changing the position of the supporting thread and hitting the coin at a 45 degree angle to the thread (see figure 7). The shift in frequency is comprised between 5 and 30 Hz. Coin Helvetia (1890) Louis-Philippe (1839) Franz-Josef (1882) Bonaparte (An XIII)
Alloy Au-Cu Au-Cu Au-Cu Au-Ag
0o orientation 5,375 Hz 5,531 Hz 5,478 Hz 5,344 Hz
45o orientation 5,379 Hz 5,563 Hz 5,495 Hz 5,350 Hz
Table 5: Fundamental frequencies f(2,0)
4.3
Testing the fakes
To train its cashiers during the period of the gold standard, the Banque de France collected samples of the most dangerous counterfeits circulating at the time. These fakes had the correct diameter and their weights were within the legal limits of tolerance. Furthermore, they all rang well. Four of these counterfeit coins were tested. Two were made of platinum and were covered in a thin gold leaf (the gold had been removed in several places revealing a grey core). These coins were the correct weight, indicating that the core was made of a metal with a comparable density to gold (excluding silver). Unfortunately, they were not documented and had only an inventory (?) number. The two other coins were thicker (between 1.5 and 2 mm) and 28 They
would imply qsq m = 1, 021
10
0 0
1 000
2 000
3 000
4 000
5 000
6 000
7 000
8 000
9 000
10 000
-20 f3= 5,326 Hz
f1= 4,240 Hz -40 f0= 2,880Hz -60
-80
-100
-120 f2= 5,060 Hz
-140 F - AN XIII (1805)
CH - 1890
AT-1881
-160
Figure 5: Frequency spectra for the Bonaparte (1805), the Helvetia (1890) and the Franz-Josef (1882) laid on a finger apparently made of a homogeneous metal29 , probably a gold-silver alloy. When pinged with a metal rod, they all produced a high-pitched resonating sound. Nevertheless, the spectrum analysis showed discrepancies with the genuine coins (see table 4.3). The platinum coins had a fundamental frequency of 6,630 Hz, consistent with the theoretical result (6,466 Hz). The gold alloy (?) coins had frequencies that were similar to those of the genuine coins because of the shape factor which was twice that of the genuine one (qs = 1.44 for the real coin and is within the range 2-3 for thick fakes). This implied a lower metal factor (low E and high nu). Nevertheless, this type of coin was probably detected because of its abnormal thickness. Coin Napoleon III with naked head (1860) G´enie (1839) Coq (1908) Louis-Philippe (1831)
Alloy Pt + Au plated Pt + Au plated Au+Ag ? Au+Ag ?
Mass 6.45 g 6.36 g 6.46 g 6.47 g
Frequency 6,320 Hz / 6,620 Hz 6,630 Hz 5,290 Hz / 6,040 Hz 5,760 Hz
Table 6: Fundamental frequencies for fakes The discrepancy between the gold and platinum coins is around 1,000 Hz.
5
Listening to the tone of gold and platinum
The human ear is sensitive in the range of 20 to 20,000 Hz with a peak at around 3,500 - 4,000 Hz and a frequency resolution of around 5 Hz under clinical conditions. 29 There
were no areas where the surface had been scratched off.
11
0 0
1 000
2 000
3 000
4 000
5 000
6 000
7 000
8 000
9 000
10 000
11 000
12 000
-20
-40
-60
-80
-100
-120
-140
-160
Figure 6: Frequency spectra for the Helvetia (1890) dropped onto a block
Figure 7: a) 0o b) 45o orientation From a musical standpoint, sounds can be ordered by their perceived pitch on a frequency scale. An octave corresponds to a doubling of frequency and comprises 12 notes. As the human ear also perceives notes in full octaves in a similar way, because of harmonics, they are given the same names (see table 5) and form a pitch class (see [20]). The chromatic scale divides the octave into 12 musical notes, each a semitone30 apart on a logarithmic scale. The fixed point is the frequency of the A4 note which is set to 440 Hz. The genuine gold coin has a fundamental frequency corresponding to an F (F8 ) while the platinum fake emits a G] (G]8 ). These two notes are 3 semitones apart31 . The two notes F8 and G]8 both belongs to the eighth octave which is the upper limit for musical instruments. The piano plays between A0 and C8 30 The
1
semitone interval is 1.059 =2 12 . See [32] corresponds to a minor third interval. These intervals are strongly consonant, with a “melancholy flavour” to the sound. 31 It
12
Note A4 A]4 B[4 B4 C5 C]5 D[5 D5 D]5 E[5 E5 F5 ] F5 G[5 G5 G]5 A[5
Frequency (Hz) 440 Hz 1 466 = 440 ×2 12 2 493 = 440 × 2 12 3 523 = 440 × 2 12 4 554 = 440 × 2 12 5 587= 440 × 2 12 6 622 = 440 × 2 12 7 659 = 440 × 2 12 8 698 = 440 × 2 12 9 739= 440 × 2 12 10 783 = 440 × 2 12 11 830 = 440 × 2 12
Note A7 A]7 B[7 B7 C8 C]8 D[8 D8 D]8 E[8 E8 F8 F]8 G[8 G8 G]8 A[8
Frequency (Hz) 3,520 Hz = 440 × 8 1 3, 729 = 3, 520 × 2 12 2 3,951 = 3, 520 × 2 12 3 4,186 = 3, 520 × 2 12 4 4,434 = 3, 520 × 2 12 5 4,698= 3, 520 × 2 12 6 4,978 = 3, 520 × 2 12 7 5,274 = 3, 520 × 2 12 8 5,587 = 3, 520 × 2 12 9 5,919= 3, 520 × 2 12 10 6,271 = 3, 520 × 2 12 11 6,644 = 3, 520 × 2 12
Table 7: Chromatic scale (see fig. 8) and the piccolo (small flute) is almost the only concert instrument which plays up to the eighth octave. The highest sopranos can barely go above 1,000 Hz (C6 ). As a result a singer cannot set a gold coin (either genuine or fake) into resonance with his voice. 0
A0
1
C1
3
2
A1
A2
5
4
A3
C4
A4
6
A5
8
7
A6
A7
8
G#8
C8 F8
Figure 8: Piano keyboard extended to the eighth octave Counterfeits can be detected musically in two ways: by listening to the sound the coin makes when rung, recognizing the pitch and labeling the note as an F or a G] , as relevant or by comparing the music made by a suspect coin with that made by a genuine one. The first method is extremely difficult and requires “perfect” or “absolute pitch”. It is estimated that less than 0.1% of the population have this ability (see [21]). It is a special gift and requires extensive training. The second method , on the other hand,(relative pitch) is easier; anyone without hearing difficulties can easily distinguish between two successive distinct tones. For instance, ringing a suspect coin three times and then a genuine coin should produce a musical sentence (see below): if the coin is genuine, the four notes should be identical, while if the coin is counterfeit the sequence should sound like the opening of Beethoven’s Fifth Symphony32 (Ta-Ta-Ta-Taa)... 32 Fate
Knocking at the gate.
13
\ˇ \ˇ \ˇ ˘ 8
G Pt Pt Au Pt As I am not fortunate enough to have absolute pitch, I tested the relative pitches of the gold and platinum coins using two different methods:1) with the recordings of the actual sound made by the genuine and counterfeit coins when rung and 2) using synthesized tones33 of 6,630 Hz and 5,400 Hz. For each method, two sets were produced: Au/Au/Au/Au and Pt/Pt/Pt/Au. I was able to distinguish the fake from the genuine coin without any difficulty.
Conclusion Two main conclusions can be drawn: the Kirchhoff theory offers a correct framework for modeling coins, and ringing coins proves to be an effective way to detect platinum forgeries which were apparently the hardest to detect. Further research should be carried out to determine the composition and alloys of the counterfeit coins. Tests should also be performed on low quality counterfeits that might be found with collectors. Another interesting point would be to test if gold bars can be pinged and if they show a different response. Obviously a different theoretical model should be developed since bars cannot be considered to be thin. Other methods could also be tested.
Acknowledgements The author wishes to thank Nathalie Boutin, Vicky Buffery, Thierry Coatantiec, Fran¸cois de Coustin, Laurent Gourier, Patrick Haas and Jean-Renaud Lefeuvre and two anonymous referees for their help and their useful comments.
33 Using
the Audacity Tone Generator function with Sine Waveform.
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References [1] George A. Akerlof, ”The Market for ’Lemons’: Quality Uncertainty and the Market Mechanism”, Quarterly Journal of Economics 84 (3): 488-500, (1970) [2] Fames Anderson and Mike Maloney, Fake Silver and Gold - How to avoid them, 2014 http://www.youtube.com/watch?v=enBD3GsIaJU and http://goldsilverworlds.com/gold-silver-price-news/detect-fakegold-silver-coins-with-an-easy-ping-test/ [3] Justine Bayley, “Medieval precious metal refining: archaeology and contemporary texts compared”. In Martinon-Torres, M. and Rehren, T. (eds) Archaeology, history and science: integrating approaches to ancient materials. Walnut Creek: Left Coast Press, 2009, pp. 131-150 [4] Laurent Jean Chaudet, L’art de l’essayeur, Desmarais, Paris, 1835 [5] F. Delamare, Study of the minting of a fake French ‘gold’ coin: A “Louis with dressed bust”, dating from 1775, Journal of Mechanical Working Technology, 11 (1985) 189-199 [6] Jacob R. Eckfeldt and William E. Du Bois, A Manual of Gold and Silver Coins of All Nations, Philadelphia, 1842. [7] Jacob R. Eckfeldt, and William E. Du Bois, New varieties of gold and silver coins, counterfeit coins and bullion, with mint values, Philadelphia, 1849 [8] Andr´e Gide, The Counterfeiters, Penguin, 1966 [9] John J. Gilman, Chemsistry and Physics of Mechanical Hardness, Wiley & Sons, 2009 [10] Jo Hermans, Physics in daily life: singing glasses, musical mugs, Europhysics new, Volume 43, Number 4, July-August 2012, p. 17 [11] M.J. Hughes and W.A. Oddy, A reappraisal of the specific gravity method for the analysis of gold alloys, Archaeometry 12 (1) (1970) 1-11 [12] Susan La Niece, Technology of Silver-Plated Coin Forgeries, in Archibald, M.M. & Cowell, M.R., Metallurgy in Numismatics, Volume 3 (1993), Royal Numismatic Society Special Publication No. 24, London [13] Charles M. Larson, Numismatic Forgery, Zyrus Press, 2004 [14] Erik Lassner and Wolf-Dieter Schubert, The History of Tungsten (parts I and II), Itia Newsletter June and December 2005, International Tungsten Industry Association [15] Leissa, Vibration of plates, NASA, vol.160, 1969 15
[16] Domini Mazzoni and Roger Dannenberg, http://audacity.sourceforge.net/about/credits
Audacity,
[17] M. Mongez, M´emoires sur l’art du monnoyage chez les anciens et chez les modernes, Imprimerie Royale, Paris, 1829 [18] John H. Munro, The Technology and Economics of Coinage Debasements in Medieval and Early Modern Europe: with special reference to the Low Countries and England, Working Paper 456, University of Toronto [19] Oddy, W.A. and M.R. Cowell, The technology of gilded coin forgeries. in Archibald, M.M. & Cowell, M.R., Metallurgy in Numismatics, Volume 3 (1993), Royal Numismatic Society Special Publication No. 24, London [20] Hendrik Purwins, Profiles of Pitch Classes: Circularity of Relative Pitch and Key - Experiments, Models, Computational Music Analysis, and Perspectives, Ph.D. Thesis. Berlin: Technische Universit¨at Berlin, (2005) http://opus4.kobv.de/opus4tuberlin/files/1109/purwins hendrik.pdf [21] Michael Ramscar, Edward Suh & Melody Dye, pitch category learning comes at a cost to lute frequency representations, Working Paper, http://psych.stanford.edu/∼michael/papers/2011 ramscar
How abso2011, pitch.pdf
[22] Chellapilla Kameswara Rao and Lokavarapu Bhaskara Rao, Vibrations of Elastically Restrained Circular Plates Resting on Partial Winkler Foundation, The Open Acoustics Journal, 2009, 2, 68-74 [23] F. Reiff, M. Bartels, M. Gastel, H. M. Ortner, Investigation of contemporary gilded forgeries of ancient coins, Fresenius J Anal Chem (2001) 371: 1146-1153 [24] R.-A. Reiss, Manuel de Police scientifique, t. 1, Payot, 1911 [25] Thomas D. Rossing and Neville H. Fletcher, Principles of Vibration and Sound, Springer, 2004 [26] Wayne G. Sayles, Classical Deception, Counterfeit, Forgeries and Reproduction of Ancient Coins, Krause Publications, 2001 [27] Thomas J. Sargent and Fran¸cois R. Velde, The Big problem of Small Change, Princeton University Press, 2002 [28] Emerson Steed, Coin identification through natural frequency analysis, Working Paper [29] Can I compute the mass of a coin based on the sound of its fall?, forum at Physics.stackexchange.com
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[30] Francois Velde, On the Evolution of Specie: Circulation and Weight Loss in 18th and 19th Century Coinage, BdF Working Paper No. 422, February 2013 [31] R.J.H. Wanhill, Embrittlement of Ancient Silver, Journal of Failure Analysis and Prevention, Volume 5(1) February 2005, DOI: 10.1361/15477020522294 [32] David Wright, Mathematics and Music, 2009, http://www.math.wustl.edu/∼wright/Math109/00Book.pdf [33] Daniel W. Zietlow, Donald C. Griffin, and Thomas R. Moore, The limitations on applying classical thin plate theory to thin annular plates clamped on the inner boundary, AIP Advances 2, 042103 (2012); doi: 10.1063/1.4757928
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