Studies in History and Philosophy of Science 40 (2009) 344–351

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Studies in History and Philosophy of Science

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Ptolemy and the meta-helikôn

Andrew BarkerInstitute of Archaeology and Antiquity, University of Birmingham, Birmingham B15 2TT, UK

a r t i c l e i n f o

Keywords:PtolemyGreek harmonicsScientific instrumentsHelikônExperimentMathematics

0039-3681/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.shpsa.2009.10.003

E-mail address: andrewqbarker@hotmail.com1 Harmonics was the study of structures such as scal

subject, which can be conveniently labelled as ‘empiricarepresented by numbers, and the intervals between thecorrelation between a musical interval and the ratio benote sounded by the full length of string is always exacFor fuller details see, for example, Barker (2007), pp. 2

2 For a thorough study of the monochord and other(2009, in press).

a b s t r a c t

In his Harmonics, Ptolemy constructs a complex set of theoretically ‘correct’ forms of musical scale, rep-resented as sequences of ratios, on the basis of mathematical principles and reasoning. But he insists thattheir credentials will not have been established until they have been submitted to the judgement of theear. They cannot be audibly instantiated with the necessary accuracy without the help of speciallydesigned instruments, which Ptolemy describes in detail, discussing the uses to which each can be putand cataloguing its limitations. The best known of these instruments is the monochord, but there are sev-eral more complex devices. This paper discusses one such instrument which is known from no othersource, ancient or modern, whose design was prompted by the geometrical construction known as thehelikôn. It has several remarkable peculiarities. I examine its design, its purposes, and the merits andshortcomings which Ptolemy attributes to it. An appendix describes an instrument I have built to Ptol-emy’s specifications (possibly the first of its kind since the second century BC), in an attempt to findout how satisfactorily such a bizarre contraption will work; and it explains how various practical prob-lems can be resolved.

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Greek treatises in the mathematical style of harmonics1 regu-larly refer to one kind of instrument that was designed for theoret-ical or ‘scientific’ purposes rather than for musical performance; itis of course the monochord. The instrument itself is rarely described;all we are usually told, for instance in the Sectio canonis of around300 BCE, or in passages quoted by Theon of Smyrna from Thrasyllusin the first century CE and Adrastus around the beginning of the sec-ond, is how to divide its string in ratios that will produce the basicconcords, or (more ambitiously) how we can construct on it a com-plete diatonic or chromatic scale. These writers give only the essen-tial mathematical data and say nothing about the instrument’smaterial construction; and there is little to encourage the idea thatany of them actually built or used one. But some people evidentlydid; in the second century CE Ptolemy discusses (with some scorn)

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es and patterns of attunement, whil’ and ‘mathematical’; only the lattem were represented as ratios of numtween relevant dimensions of a soutly an octave below the note sound5–29.instruments used in a Greek musica

the attempts of a rather earlier theorist, a certain Didymus, to makethe monochord easier to use effectively in practice, and his accountpresupposes that others before Didymus had also made use of thereal thing, not just of diagrams and formulae (Harm. II.13).2

The real, material instrument was certainly of interest to Ptol-emy himself, and he describes it at considerable length (Harm.I.7), commenting in particular on issues that are irrelevant to itsabstract, mathematical credentials, but plainly need to be ad-dressed if it is to be built and used. He explains very carefully,for instance, how we can test that the whole length of the stringwe are using is evenly constituted. He discusses the design of thebridges, which for mathematical purposes are represented merelyby straight lines, and recommends using bridges whose upper sur-faces are segments of spheres. He points out that the movable

ch form the framework for musical melodies. There were two main approaches to ther concerns us here. In this form of the science, notes were conceived as quantities andbers. The original inspiration for this approach came from observation of the regular

nding body; on a stretched string of consistent thickness and tension, for instance, theed by the half-length. Hence in mathematical harmonics the ratio of the octave is 2:1.

l scientist’s ‘laboratory’, including the instruments discussed in this paper, see Creese

Fig. 1. The helikôn. (Illustrated by the author.)

Fig. 2. The ‘meta-helikôn’. (Illustrated by the author.)

A. Barker / Studies in History and Philosophy of Science 40 (2009) 344–351 345

bridge which is shifted to different positions to produce the variousratios must be a little higher than the fixed bridges at the ends ofthe instrument, in order to ensure firm contact with the string;and since this seems mathematically scandalous he argues quiteelaborately, in Harm. III.2, that although the string will not in prac-tice form the straight line that represents it in a diagram, this willnot distort the mathematical relations between the lengths of itssections if proper precautions are taken. There is a good deal morein the same vein. His declared methodology for the science de-mands, in fact, that such instruments should actually be builtand used, since he insists on a procedure which first derives thesets of ratios defining divisions of the tetrachord from mathemat-ical principles, and then uses instruments of appropriate sorts tosubmit the derived results to the judgement of the ear. This pointsto an authentically experimental approach, quite unlike that of ear-lier theorists; when they used the instruments at all, they seem tohave done so merely to present their constructions in audible form,not with any intention of putting the conclusions of their theoris-ing at risk. They typically took the view that perception does nothave the competence to pass judgement on conclusions reachedby rational means. I have argued elsewhere that Ptolemy meanswhat he says; there are solid indications that he had indeed fol-lowed his own prescriptions, and he gives his readers all the helpthey need if they are to make their own instruments and conductthe relevant tests for themselves.3

Now in Ptolemy’s opinion, the monochord’s usefulness is lim-ited. It can indeed be used to assess the ratios assigned to the sim-ple concords, but because of the delays and difficulties involved inmoving its bridge to a succession of different, precisely demarcatedpositions, it cannot provide the basis for reliable judgements aboutthe credentials of whole scale-systems (Harm. III.12; I shall say alittle more about the reasons later). For those purposes more com-plex instruments are needed, and he describes several sorts, againincluding a good many details which are relevant only if they are tobe constructed and deployed in practice. In the rest of this paper Ishall be discussing just one of them, or rather, as I shall try to ex-plain, one rather special and intriguing way in which at least one ofthe more elaborate instruments can be designed and used. The ba-sic discussion is in Harm. II.2, and there are additional comments inII.16, III.1 and III.2.

The starting point in II.2 is a construction called the helikôn,which was devised to demonstrate the ratios of the concords, Ptol-emy tells us, by hoi apo tôn mathêmatôn. I take this phrase to mean‘specialists in mathematics’, and the way in which their procedureis described points strongly to the conclusion that they were

3 Barker (2000), pp. 192–258.

simply constructing a geometrical diagram (Fig. 1). Lines in the dia-gram can certainly be interpreted as strings, bridges and so on, asPtolemy explains in a later paragraph, but the initial manner ofpresentation is plainly that of an exercise in geometry. The firstsentence will give the flavour: ‘They set out a square, ABCD, andafter dividing AB and BC in half at E and F they draw lines joiningAF and BGD, and draw EHI through E and JGK through G, parallel toAD’. The point of the diagram is that if we work out the ratios be-tween lengths of the vertical lines divided off by AF they will turnout to be the ratios of the concords. Thus, for instance, AD stands toBF and FC in the ratio 2:1, the ratio of the octave, to EH in the ratio3:1, the octave plus fifth, to HI in the ratio 4:3, the fourth, and to GKin the ratio 3:2, the fifth. I shall not catalogue the entire set of suchrelations; suffice it to say that all the ratios of concords in the spanof a double octave are represented, most of them several timesover. The technique used to show all these relations relies on famil-iar facts about similar triangles. The conclusions can be presentedaudibly if we treat the vertical lines as strings, all initially tuned tothe same pitch, and line AHGF as a continuous bridge; but Ptol-emy’s remarks to this effect come in a separate passage, wherehe is probably no longer relying on the mathêmatikoi responsiblefor the diagram. (I should point out that we need not consider diag-onal BD at this stage, since it is only a line of construction and hasno material counterpart.)

All that this gives us directly is an ingenious way of constructingthe ratios of concords. But somebody—and I suspect it was Ptolemyhimself—saw how the principles governing it can be applied in an-other way, as the basis of a much more promising and versatileconstruction. He does not give it a name; the cumbersome designa-tion ‘meta-helikôn’ in the title of this paper is my own rather feeblecoinage. In this case (Fig. 2) ABCD is a rectangle whose proportionsare mathematically irrelevant. Its base, DC, is extended as far as E,where DC and CE are equal, and the line representing the bridgenow runs from T to F, cutting BC at its mid-point, F. If we nextadd other vertical lines representing strings, such as GKJ and HLI,they can be placed anywhere we choose so long as they are parallelto AD and BC. Wherever we put them, the ratio between thelengths lying below the bridge, in this case KJ and LI, will be thesame as the ratio between the lengths along the bottom of the dia-gram from T to J and from T to I. We can prove it once again by ap-peal to straightforward facts about similar triangles. And of coursewe can put in as many more strings as we like, wherever we like solong as they run parallel to AD, and the same will be true of them.

It should therefore be possible to build an instrument with en-ough strings to produce a complete scale, and to adjust theirsounding-lengths to ratios chosen on theoretical grounds by locat-ing them at appropriate distances from T. As a preliminary, weneed to prepare a measuring strip or ruler—Ptolemy calls it akanonion—equal in length to DE, marked off into 120 equal seg-ments; this is to be attached to the instrument alongside DE. We

346 A. Barker / Studies in History and Philosophy of Science 40 (2009) 344–351

can then pick out on it (perhaps in different colours) the distancesfrom E that stand in the ratios appropriate to each of the scales weare interested in. Here Ptolemy has done the hard work for us; thetables in II.15 conveniently identify for us all the relevant positionson the 120-unit length.4 In order to try out the designated ratios forany of our scales, we simply slide the strings along to the appropriatemarks. Further, as Ptolemy points out, the bridge, FT, does not needto begin at F and to cut BC at the mid-point or in any other particularplace; everything will still work if we shift its top end downwards,pivoting on T (for instance into the position indicated by the dottedline EM). The relations between the strings’ sounding-lengths will beexactly the same; all that will happen is that the system is trans-posed upwards in pitch, into what we would call a different key.

This is all very well in theory, but is it possible to build a realinstrument which would give reliable results, and on which thesemanipulations would not be too appallingly cumbersome? Ptol-emy takes great pains to show that it is. I am convinced, in fact,that he had one made to his specifications, and that it did indeedwork as he says, though there were a few practical problems tobe overcome, as we shall see. The first indication is in the orderin which he describes the stages of the instrument’s construction.The sequence of steps in the mathematicians’ construction of thehelikôn is exactly what one would expect in a textbook of geome-try, but from that point of view the order in which the steps of thissecond construction are set out is frankly chaotic. There must besome reason for that. Ptolemy was an accomplished mathemati-cian, and his deliberately mathematical arguments elsewhere inthe Harmonics are impeccably organised. Again, whereas in the ac-count of the helikôn Ptolemy gives the complete mathematicaldescription first, and only then identifies specific lines as corre-sponding to strings, bridges and so on, in the second case thesematerial features are specified right from the start; thus AD andBC are identified in the opening sentence as vibrating strings,and as giving the highest and lowest notes of the octave, whichtells us that all the other strings will be placed between them.We are told to tune all the strings to the same pitch even beforethe construction is complete; only afterwards are we told aboutthe line joining F to T. But the order makes perfect sense if wethink of the passage as a page from a manual giving practicalinstructions for building instruments of this sort. It is indeed math-ematically haphazard, but if a craftsman setting out to make thedevice conscientiously followed the order corresponding to thesteps in the construction of the helikôn, he would find himself hav-ing to remove bits of the instrument that he had already put inplace in order to fit other parts later. In particular, he evidentlymust tune the strings to equal pitches before adding the bridgeindicated by the line from F to E; otherwise he would have to re-move it again in order to do the tuning. Nor would any useful pur-pose be served by setting out the diagram geometrically andexplaining the underlying mathematics before going on to explainwhich lines correspond to which material components. A crafts-man needs to know from the start what the various lines represent.

Ptolemy is telling us, then, how the instrument should be builtin practice. But the instructions in II.2 leave one glaring difficultyunaddressed. How are we to move the strings across the face ofthe instrument without all the complications involved in unstring-ing it, removing the diagonal bridge, screwing tuning-pins intonew positions (which would be almost impossible to do accu-rately), re-fitting the strings and re-tuning them, replacing thebridge and so forth? Ptolemy does not tackle the issue until four-teen chapters later, in II.16. Here he is not thinking initially aboutthe device we are interested in, but of the straightforward

4 Almost all of them in fact lie in the range between 60 and 120. In a few of the syscorresponding string-length made shorter; the smallest number involved is just over 56. Athe 120 mark; the largest number needed is approximately 124½.

eight-stringed instrument described in I.11; it is effectively (asPtolemy says himself) a monochord which has been kitted out withseven additional strings, and each string has a movable bridge of itsown. He now tells us that it will be helpful to fix both ends of eachstring to kollaboi, tuning-pins, rather than having tuning-pins atone end only and fixing the other immovably in place. This is sothat a substantial length of each string can be wound round onepin and then gradually unwound from it onto the other, so as tofacilitate the process, described in I.8, by which we check thatthe string is evenly constituted all along the length of it that weshall use. He then adds that it will be useful to make all the tun-ing-pins capable of moving in (or on) something called the pelekê-sis, in such a way that they can slide along the sides of theinstrument. This, he says, is ‘for the sake of the second form ofusage, in which a single flat bridge is placed under the strings,and the sideways movements of the strings make the appropriateattunements’. This ‘second form of usage’ is plainly the system weare considering. If the tuning-pins stay rigidly in the same placeand the strings are simply pulled sideways and somehow jammedin a new position, their tension and hence their pitches will be al-tered, and we shall have to remove the angled bridge and re-tunethem to equalise their pitches again. If we can slide the wholestring across the face of the instrument, complete with its tun-ing-pins, this problem will be neatly resolved. (I discuss ways inwhich this can be done in the Appendix to this paper.) Ptolemy rec-ommends further that we mark out two kanones or rulers in iden-tical ways, and fix one of them alongside each of the two fixedbridges, that is, in our diagram, alongside AB and DC; we can thenline up both ends of each string against the corresponding markson the two kanones, and be sure that the strings are still at a rightangle to the fixed bridges. That too seems eminently sensible.

When these additional features are taken into account, it shouldbe possible to build such an instrument, and it should function inthe way that Ptolemy explains; I describe in the Appendix the work-ing model I have built myself. But it is worth noticing that in thispassage Ptolemy does not refer to the device as an instrument. Aftera couple of sentences about the tuning-pins on the eight-stringedinstrument directly derived from the monochord, he does not say‘Now let us consider another kind of instrument’, but goes on inthe same sentence to say that it will also be useful to make them,those same tuning-pins, movable in the pelekêsis. He says that thisis ‘for the sake of the second form of usage’, tês heteras tôn chrêseôn,not ‘for use on another kind of instrument’; and if we now look backat the original description in II.2 we can see that he uses an expres-sion there which could carry much the same meaning. When com-paring this system with the one in which a separate movable bridgeis placed under each string, he talks about a first and a second ‘man-ner’ or ‘style’, tropos, and not about two different instruments. Herehe does not say whether they are styles of usage or styles of instru-ment, but the latter is lingistically less likely; a tropos is usually away of doing something, rather than a species of object (for whichone would expect a noun such as eidos); and the probability that hemeans the former is increased when he returns to these matters inIII.1. In that passage the terms that have appeared separately inthese contexts in Book II, chrêsis (‘usage’) in II.2 and tropos (‘manner’or ‘style’) in II.16, come together in the phrase ‘manner or style ofusage’ (tropos tês chrêseôs).

All this makes perfectly good sense. The fact that the octachordinstrument’s strings are mounted in such a way as to make themmovable does not entail that the executant must always actuallymove them when he is setting up an attunement. Sometimes itmay be more convenient to use the other method, putting a

tems of attunement described, the highest note needs to be raised slightly and thefew of the systems, similarly, require notes lower than the one produced by a string at

Fig. 3. Two octaves on the ‘meta-helikôn’. Strings 1–7 are initially tuned an octavebelow strings 8–15. The part of the bridge HE that runs under the strings of theupper octave can be made a little lower than the rest. String 8 serves both as thehighest note of the lower octave and as the lowest note of the higher. (Illustrated bythe author.)

A. Barker / Studies in History and Philosophy of Science 40 (2009) 344–351 347

separate movable bridge under each string in the usual way; andthere are three passages, in II.2, III.1 and III.2, where Ptolemy ex-plains that both forms of usage have advantages and disadvan-tages, or identifies situations in which one or the other of themis unusable. It will therefore be helpful to have both methods avail-able, and one can use the same instrument for both of them, solong the strings can be moved.

But why should this rather peculiar approach should be thoughtnecessary or even helpful? Can it achieve anything that is impossi-ble or more difficult or less reliable when we simply give eachstring a small movable bridge of its own? The three most obviouslyrelevant passages in Ptolemy’s text give us no clear answer. In II.2he considers which of the two methods is easier to use in practice,and he seems to give them roughly equal ratings. What he calls the‘first’ method, the one with a separate movable bridge for eachstring, has the advantage that one does not need to shift the stringssideways, but the disadvantage that in order to create a newattunement one has to manoeuvre several fiddly little bridges intothe correct positions. We might add that according to the instruc-tions that Ptolemy has given, when we do this we have to align anappropriately marked ruler, which is a separate piece of equipmentand not part of the instrument, against each of the strings in turn inorder to find the locations of the bridges. This will obviously betime-consuming and errors can easily creep in. We have to holdthe ruler absolutely still in exactly the right position while adjust-ing each bridge; we must place each bridge exactly under the rightmark on the ruler; and we have to make sure that all the bridgesare always perfectly vertical, so that the angle at which each stringis bent out of the straight by its bridge is the same. Conversely, thesecond system, the one in which the strings’ tuning-pins slide side-ways in the pelekêsis, gets a good mark for avoiding the problemsassociated with shifting a number of small bridges, but a bad markbecause it requires us to move the strings themselves. On thewhole, it seems to me, this latter system’s overall score shouldbe rather the better by these criteria; the two rulers that areneeded are attached to the instrument, so that one need not worryabout getting them in the right position each time and holdingthem precariously in place while making the adjustments. Weneed only make sure that each string runs directly over the rele-vant marks, and that it has not been bent out of the true at thepoints where it crosses the bridges. Ptolemy, however, makestwo other points about the second method in this passage whichhe apparently thinks relevant. On the plus side he puts the fact thatit is possible to shift the whole attunement to a different range ofpitch while preserving its interval-structure, simply by pivotingthe bridge on point E; and on the minus side, he puts the fact thatthe distances between the strings when an attunement has beenset up will never be equal, as they can be when we use the othermethod. Sometimes, Ptolemy says, they will differ by largeamounts. But he does not explain why this is a disadvantage, orwhy we should welcome the possibility of transposing an attune-ment to a different pitch. I shall come back to these problems later.

III.1 and III.2 are quite dauntingly complex. We need not pursueall their intricacies in order to understand the comments that arerelevant here, but we do need to put them in context In III.1 Ptol-emy is discussing the best way of setting up an instrument with fif-teen strings, to accommodate the whole range of the two-octaveperfect system. He suggests that in order to avoid the loss of sonor-ity that would be caused by making the lengths of the strings forthe highest notes very short, we should use thinner strings forthe eight notes of the upper octave; and when we tune thesestrings to equal pitches before inserting the bridges we should

5 The general principle underlying these remarks (but with none of the details set out byon Plato’s Timaeus quoted at Theo Smyrn. 61.20–23 Hiller (where in lines 22–23 I read pr

make them an octave higher than those in the lower range. Then,since the interval-structures of the upper and lower octaves inthe perfect system are always the same, we only need to workout the division of our ruler for a single octave, and apply it to eachin turn. For those purposes, he says, we need only the ability tojudge whether the strings’ pitches have been correctly equalisedat the start; but if we have the skill to recognise accurately, byear, the right interval-patterns for each division, we can also workthe other way round. That is, we can start with the strings tuned toany pitches whatever, place the bridges (or if we are using the slid-ing system, the strings) in the mathematically correct positions,and then tune the strings to the appropriate pitches by ear. If wedo that, and if the mathematical derivations are reliable, we shallfind that when we move the bridges or slide the strings into thepositions which the mathematics assigns to another pattern ofattunement, we shall again produce the right results, and that ifthe movable bridges are removed altogether, all the relevantstrings will turn out to be tuned to equal pitches.5

There is a puzzle here; if all we can do is to recognise when thestrings are equal in pitch, how are we supposed to judge whetherthe results produced by the application of the ratios are musicallycorrect? I shall come back to that issue too; but first let us considerthe problem he identifies in using the sliding system in this con-nection. Two sets of strings and two bridges will be needed, onefor the lower octave and one for the upper. But then, he says, ‘it willoften happen that in the sideways movements involved in shifts oftuning, the strings located by the ends of the bridges, in the middleof the span of the kanôn, come up against the ends of the bridgeslying opposite to them, and so can no longer maintain their properlengths. Hence it is possible by this method to determine onlythose systems in which one or other of the notes mentioned keepsthe same position in the shifts of tuning’. A glance at Figure 3 willshow that difficulties of this sort might indeed occur, most obvi-ously if the bridges have been pivoted downwards to raise theoverall pitch. If the right-hand bridge, GF, were pivoted down-wards it might get in the way of string 7 in the left-hand set, espe-cially if the attunement demands that this string be moved to theright (cf. n. 3 above); and if string 8 is moved to the left it maycome up against the part of bridge HE which has not been madelower (enough of HE’s length must be at full height to allow it tocome in contact with string 7 when it is pivoted downwards).We may suppose that the problem could easily be solved, simplyby leaving a wider space between the two sets of strings. The rea-son why Ptolemy does not mention this possibility might bemerely that the instrument he was actually using did not have

Ptolemy) is stated briefly in a rather earlier text, a passage from Adrastus’ commentaryolêphtheisêi for the MSS proslêphtheisêi).

348 A. Barker / Studies in History and Philosophy of Science 40 (2009) 344–351

enough room to allow it. But another interpretation is possible, andthat too is an issue to which I shall return.

We can deal fairly quickly with the passage in III.2. Here Ptol-emy is describing a way of getting all the notes of a two-octave sys-tem out of only eight strings. When we are using the methodwhere separate movable bridges are placed under each string, thisis done by having two such bridges for each, and adjusting theirpositions so as to form one octave of the attunement at one endof the strings, the other at the opposite end. Once again, however,there is the problem of inadequate sonority when the lengths getvery short. We should therefore divide the eight strings into twogroups of four, and tune one group to a pitch which is either a per-fect fifth or a whole octave above the other. We then construct theintervals of the lower octave on one set of four strings and those ofthe higher octave on the other, producing two notes on each stringby using the distances from the fixed bridges at each of its ends tothe nearer of the two movable bridges that divide it. Thus we useone end of each of the four lower-pitched strings for the first fournotes of the lower octave and the other end for the notes from thefifth to the eighth, and extract the whole upper octave out of thehigher-pitched strings in the same way (the highest note includedin the lower set will be the same as the lowest in the higher set).

But this, as Ptolemy points out, is impossible by the methodwhich slides the strings sideways. This is because the ratios be-tween the sounding-lengths needed at one end of a set of stringswill usually be different from those needed at the other. That posesno problems for the method in which each string has its own pairof small movable bridges. On the sliding system, however, the ra-tios correspond to the lateral distances between pairs of strings.Each string must be moved as a whole, and the distance betweenany string and its neighbour must always be the same at both ends.Hence the trick cannot be done. If we want to set up our attune-ments in this way, we cannot use the sliding method.

Why then does Ptolemy think this system worth bothering withat all? One of the two advantages he has found in it seems rathertrivial—that is, the fact that we do not have to manipulate a num-ber of small bridges—and he has not explained why the other, thatwe can easily transpose the whole attunement to a different pitch,should be thought of as an advantage at all. We also need to askwhy he thinks the unequal distances between strings problematic,why he does not envisage setting the two sets of strings describedin III.1 far enough apart to prevent the string at the end of one setfrom impinging on the bridge belonging to the other, and how heimagines that someone whose ear is only good enough to recognisewhen two notes are in unison can possibly assess the credentials ofan intricate scalar structure.

Now when Ptolemy suggests in II.2 that it is helpful to be able totranspose a whole system to a different pitch, the word he uses for‘pitch’ is tonos. Tonos can indeed mean simply ‘pitch’, but systemscalled tonoi are the subject of a long series of chapters in Book II(part of Chapter 6, and the whole of Chapters 7–11). In this contextthe word is often translated as ‘keys’, and the translation is notseriously misleading. A little later, in II.15, we are faced with anarray of fourteen five-column tables of numbers specifying differ-ent divisions of the octave, and these, from one perspective, arethe climax of the Harmonics. The five columns in each table repre-sent sequences in different genera or combinations of genera; butthe tables themselves are distinguished by their tonoi, and by thenote from which the sequences represented in the table begin:‘Hypophrygian [sc. tonos] from nêtê’, ‘Lydian from mesê’, and soon. Evidently the tonoi are important, and if we are to test the accu-racy of the tables on our experimental instrument we need to beable to move from one to another. We might be inclined to assume,

6 Some arguments to support these contentions will be found in Barker (2000), 255–25

then, that the role of the pivoting bridge is to make these modula-tions in a straightforward way.

But the matter is not as simple as that. Ptolemy insists, withconsiderable emphasis, that the tonoi must not be treated merelyas transposition-keys. We can indeed think of a modulation of to-nos as shifting the established scale-structure upwards or down-wards, but its principal function is to replace the pattern ofintervals which had occupied a given range of pitch with a differ-ent pattern, shifted into this range from a different part of thescale. Thus if we modulate, in modern terms, from C major to Gmajor, the octave of pitch which was originally occupied by theinterval-sequence starting on the first degree of a major scale isnow occupied by a different sequence, the one that starts on itsfourth degree. Hence when Ptolemy is thinking about shifts oftonos, the effect he has in mind and which is represented in thetables of II.15 cannot be produced by straightforward transposi-tions, like those that can be achieved by pivoting the bridge ofour instrument. We need to keep the pitch-range more or less un-changed, and project onto it a new arrangement of intervals. Wecannot do that just by pivoting the bridge.

Now the chapter which introduces the procedure of sliding thestrings sideways comes immediately after two others, I.16 and II.1,where we find some surprising developments. In I.15 Ptolemy hasset out six divisions of the tetrachord which he has derived math-ematically from first principles, each of which will form the basis ofan attunement over the span of an octave; and he has announcedthat if these are represented accurately on an eight-stringed instru-ment, the listener will find that each octave is attuned ‘so accu-rately that the most musical of men would not alter it any more,even a little’. At the beginning of I.16, however, it turns out thattwo of the six attunements will not in fact be readily recognisedby the ear, since contemporary musicians never use them, andthe harmonic scientist will therefore have no experience to guidehis perceptual judgement. The question how ‘the most musical ofmen’ would in that case assess their accuracy is never addressed.The chapter adds several other complications too, but we canignore all of them except one. Ptolemy reveals that by no meansall the patterns of attunement in regular use by musicians directlymatch any of those that emerge from applications of his theoreticalprinciples. All six of those used by lyre-players and kithara-players,which he now describes, have tetrachords that match structuresfound among his theoretically derived divisions, but three of thesix involve combinations of tetrachords of two different types,and only one type, in fact, is ever used by itself.

This retreat from theoretical purity would be surprising enoughin a treatise in mathematical harmonics even if it came in a tangen-tial parenthesis, and there is nothing like it in any other Greeksource. But as we read further it becomes clear that so far frombeing theoretically embarrassing but essentially irrelevant, theattunements used by practical musicians are in fact Ptolemy’s prin-cipal target. He discusses them again in II.1, and completes hisdescription of them in II.16, where each is assigned to its propertonos; and I am convinced that the main purpose of the elaboratetables in II.15 is to provide the information we need in order toset each of them up and test them on our laboratory instrument.It is his mathematical analyses of these attunements that must ulti-mately be brought to the judgement of the ear, and his theory willstand or fall by the results of these tests.6

If these conclusions are on the right lines, we should expect thesliding-string method to show its merits, somehow or other, inconnection with these theoretically mongrel but authentically mu-sical attunements. That hypothesis is encouraged by the locationsof the points in the text where Ptolemy discusses the method.

8.

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The main description, in II.2, comes immediately after two chap-ters on the musicians’ attunements. The next allusion, where weare told about tuning-pins moving in the pelekêsis, is in II.16, whereit runs on seamlessly from the paragraph in which the descriptionof those attunements is completed. The remaining two referencesto the method come in the chapters immediately following II.16(i.e. III.1 and III.2), and these are wholly devoted to details of theinstruments that will be used for the tests.

How, then, will the method make itself useful in this connec-tion? I invite readers to imagine that they are Greek students ofharmonic theory in second-century Alexandria, that they have readPtolemy’s treatise and are anxious to check the accuracy of his con-clusions. Imagine further that although you are interested in musicand have listened to a good deal of it, you are not professionallytrained in the art, and perhaps do not altogether trust your unaidedear when it comes to making precise discriminations betweenintervals that differ by only a very small amount. Your task, as Ihave construed it, is to assess Ptolemy’s mathematical analysesof the attunements used by the professional kitharists and lyre-players of his day. So how do you set about it? The most obviousand straightforward approach, as it seems to me, is to go and finda well qualified professional string-player, taking with you aneight-stringed instrument of the sort that Ptolemy describes. Ptol-emy has identified each of the attunements he analyses by thename which musicians use for it; so you can ask your professionalto tune his instrument to one of these, just as he would in the con-cert-hall. Meanwhile you tune the strings on your own piece ofgadgetry to the ratios that Ptolemy’s theory assigns to that attune-ment; and then you compare the professional’s results with yourown to see if they are the same.

This is all very well, but you are going to find the comparisonvery difficult to make if your instrument and his are tuned to dif-ferent overall ranges of pitch. It is at this point that the pivotingbridge will come in handy. You set up your attunement, and thenshift the bridge carefully to the angle at which one of the stringsof your instrument is perfectly in unison with the correspondingstring on his. Then, if the Ptolemaic analysis is correct, the pitchesof all your other strings will also be identical with their counter-parts on the lyre or kithara, and if they are not you have reasonto believe that the analysis is flawed. All this is hypothetical, ofcourse. But it would resolve one of the problems the text poses,by explaining how a listener who cannot trust his ear to identifyaccurately any musical relation except the unison can neverthelessjudge the credentials of a mathematically specified attunement. Ifhe goes through the steps I have suggested he can do that readily,since putative unisons are the only relations he is called on to as-sess at any stage of the process.

But why does Ptolemy give the method a black mark becausethe distances between the strings will always be unequal, and willalso change as we move from one attunement to another? This fea-ture is theoretically irrelevant, and it will pose no significant prac-tical problems either, if all we do with the device is to checkindividual notes against those produced by a professional, or listento individual intervals and assess them by ear, or play the scale wehave constructed from end to end while listening attentively to itsmusical effect. But the point, I think, is that this is by no means allwe should be doing. This comes out most clearly in two chapterstowards the end of Book II. In II.12 Ptolemy examines some ofthe limitations of the simple monochord, and in II.13 he reviewsthe rather marginal improvements devised by his predecessor Did-ymus. It comes as no surprise that all the difficulties affecting theinstrument that Ptolemy mentions are practical ones; its mathe-matical credentials are impeccable. What is likely to make thereader blink, and certainly surprised me when I first read the pas-sage, is that none of the difficulties mentioned here will arise un-less we are treating the monochord as a performing instrument,

one on which real pieces of music are to be played, and playedproperly, in the tempo, rhythm and style appropriate to them. InII.12 Ptolemy has an extensive riff on all the musically significantthings that one cannot do with it—playing fast and accurately atthe same time, playing legato, playing two notes simultaneously,shifting smoothly between notes without producing an unwantedglissando, and so on and so forth. ‘So far as practical usage is con-cerned’, he says, ‘this instrument would be the last and feeblest’.

But why should any of this worry a harmonic theorist, who isnot planning to take the instrument into the concert-hall, only touse it for testing the credentials of mathematically designedscale-systems? We cannot dismiss the passage as an irrelevantaside; Ptolemy is not given to parenthetical deviations from histheme, certainly not at this length. More importantly, at the pointin the text where these chapters appear he is working up to thestage where his constructions will be put to the test, and is movingtowards his discussions, in II.16, III.1 and III.2, of the features whichthe instruments to be used will need if they are to play their partreliably. The passages on the monochord fall into place as a preli-minary to those discussions, showing why it does not have thequalifications needed for the task.

In that case he must be implying that the experimenter willneed to play real tunes on his instruments, not just scales or selec-tions of intervals, and that in doing so he should aim to play themas they would be played in performance. But why? No other Greektheorist suggests anything of the sort; but anyone other than a pro-fessional who has tried to tune a moderately complex instrumentaccurately will probably see the point at once. I have attemptedthe feat fairly often myself, and I think my musical ear is tolerablygood. But I have repeatedly found that if I tune the instrument byear to what sounds like the right pattern of relations when I play ascale, and then try playing a familiar tune on it, complete with thetempo, rhythm and so on that we would expect, I can immediatelytell that some of the intervals are wrong. If my experience is typi-cal, some inaccuracies in an attunement will only become apparentwhen it is put to use for the musical purposes for which it isintended.

Ptolemy does not make these points explicitly, but we canhardly make sense of what he says about the monochord and aboutDidymus’ techniques unless we assume that he recognised theirforce, and that Didymus did so before him. If he did, there willbe a persuasive explanation for his apparent dislike of an instru-ment on which the strings are separated by different and variabledistances, since such a thing would be difficult to play with any flu-ency. It would be rather like trying to play a piano with keys of halfa dozen different widths, some of which became wider and othersnarrower whenever the music modulated; even a skilful musicianwould be likely to have difficulties with it. And when Ptolemy dis-cusses an instrument with two sets of eight strings tuned an octaveapart, considerations of the same sort will explain why he does notenvisage separating the two sets by enough space to prevent astring running into the end of a bridge; the gap would be unusuallywide, and it would be a tricky business to play smoothly andswiftly across the junction between the two octaves.

I said earlier that the apparatus I have been discussing wasquite probably Ptolemy’s own invention; and one reason for hisevident interest in it, despite its disadvantages, may well have beena simple delight in its—and his own—ingenuity. But he finds realmerits in it too, as we have seen, as well as the problematic fea-tures we have just been considering, and I have tried to unravelwhat these merits and defects amount to. Posterity seems to havejudged that its disadvantages outweighed its positive qualities. NoGreek theorist ever mentions it again, though the helikôn whichapparently inspired its design makes another brief appearance acentury or so later in Aristides Quintilianus De musica III.3. Nordo I know of any medieval, renaissance or later texts that refer to

Fig. 4. The working model. To simplify the drawing I show only two strings.(Illustrated by the author.)

350 A. Barker / Studies in History and Philosophy of Science 40 (2009) 344–351

it. So far as I am aware, the working model described in the Appen-dix is the first such instrument to be constructed since Ptolemy’stime (perhaps only the second that has ever existed), and when Idemonstrated it to an audience at the Whipple Museum in Cam-bridge in May 2008 it was being used for the first time in over eigh-teen hundred years.

Appendix : a working model of the meta-helikôn

The instrument I have built is not a masterpiece of the instru-ment-maker’s art; it could easily be replicated by anyone with a lit-tle experience of rough carpentry. All its components apart fromthe strings and tuning-pins came from an ordinary chain-storesupplying materials for DIY enthusiasts; the strings are nylon gui-tar-strings, and the simple metal tuning-pins, not unlike small ver-sions of those of a piano, are the kind used on folk-instrumentssuch as zithers and psalteries. Many music-shops can providethem, together with the key needed to turn them when the stringsare being tuned. The basis of the instrument is a rectangular box,approximately 95 cm long, 42 cm wide and 4.3 cm deep.7 Thetop and bottom are sheets of plywood about 0.5 cm thick, and thesides are lengths of pine with a height of 3.2 cm and a thickness of1.8 cm. For the fixed bridges I used narrow hardwood battens witha triangular cross-section, about 1 cm high (I shall explain the con-struction of the movable bridge later), and the 120-unit scales aremarked on thin pinewood strips, glued to the top of the box (thesound-board) alongside the inner edge of each of the fixed bridges.8

The completed instrument is illustrated in Figure 4; some of its lessself-explanatory features will be discussed below.

The main problem that faces a craftsman setting out on thisconstruction, one which Ptolemy’s text does not completely re-solve, is to work out a viable way of mounting the strings whichwill allow them to be moved sideways across the face of the instru-ment. I had originally thought of setting the tuning-pins in smallblocks of wood fitted into slots in the soundboard; but althoughthis would be possible, it would take some ingenuity to make itwork well, and would, I suspect, have stretched my wood-working

7 The bigger the instrument is, within reason, the better it will work. With greater widthgreater length allows the units of the 120-unit scale to be subdivided more accurately (ancumbersome fractions). The size I chose is a compromise between the requirements of ef

8 The measuring-strips are needed only along the part of the instrument that is crossecorresponding to the distance between 60 and 120 units, ignoring those of Ptolemy’s attuhappens, it turned out to be convenient to mark out the measuring-strips in imperial units.room to indicate the half-units too. Where Ptolemy’s attunements call for other fractions

skills beyond breaking-point. A better solution was suggested bymembers of the audience when I spoke about the instrument ata meeting of the British Society for the History of Mathematics inDecember 2007, before I had started to build one of my own. It isthat the tuning-pins for each string should be mounted near theends of a continuous batten or rod which runs from side to sideof the instrument, under the soundboard, and projects far enoughat each end to allow the tuning-pins to be set into it. I give a fewmore details of this part of the construction in the next paragraph,but in essence it is simple and practical, and it has the additionaladvantage that the point at which the string leaves its tuning-pincan be lower than the top of the fixed bridges, so that the contactbetween the string and the bridge will be satisfactorily firm.

If the strings and the rods or battens on which they aremounted are to be shifted sideways, the sides of the box must ofcourse be left open along the part of it that lies under the strings.The remaining part of the box has solid sides, as do its ends, andthe stability of the construction is not endangered by the presenceof these apertures. For the purposes of this working model, I usedwooden battens as the mounts for the strings. They have to bequite narrow, since in some tunings the strings must be placedclose together; and unless the instrument is very long, thicker bat-tens will impinge on one another while the strings are still too farapart. The battens I used measure 1.2 cm across the surface intowhich the tuning-pins are fixed, and this is just narrow enough,on an instrument of this length, to accommodate most (not quiteall) of the types of attunement that Ptolemy describes. Metal rodswould probably be better, since they could be made narrowerwithout becoming too weak (though there is of course a limit; theymust be at least a little wider than the tuning-pins, whose diame-ter is about 0.3 cm), but I do not have the skills or the tools thatwould be needed to fix the tuning-pins into a metal base whilemaking them properly adjustable. In order to reduce friction andmake it easier to move the battens smoothly, I glued three narrowlengths of hardwood under the soundboard to act as runners, onenear each edge and one along the centre, so that the battens wouldbe in contact only with them and not with the soundboard itself.This strategy turned out to work well.

Once the box has been constructed, the fixed bridges, measur-ing strips, battens, strings and so on have been put in place andthe strings have been tuned in unison, the remaining task is to in-sert the pivoting, diagonal bridge. There is a minor difficulty here.As Ptolemy explains, it needs to be slightly higher than the twofixed bridges. But I found that the business of inserting a higherbridge under the whole gamut of strings, once they are under ten-sion, involves some awkwardness; and one has to take great carenot to disturb the initial tuning. I solved the problem by makingthis bridge in two parts and inserting it in two stages. The first stepis to slide under the strings a bridge whose cross-section is identi-cal with that of the fixed bridges; the second is to slide a thin stripof wood of the same length between it and the soundboard, so rais-ing the height by a small amount without interfering with thestrings themselves. I had previously drilled a small hole verticallythrough a point near one end of both the bridge and the strip,and a corresponding hole in the soundboard at the point on whichthe bridge will pivot. Once the bridge and the strip are in theirplaces, I simply push a pin through their holes and into the holein the soundboard, and we have our pivot. When the bridge is

the sounding-lengths of the strings can be made longer, which improves the sound;d the numbers which Ptolemy assigns to the string-lengths sometimes involve quite

ficient working on the one hand and portability on the other.d by the strings. For the purposes of this model, I fixed them only along the stretchnements that require reference to greater or smaller numbers (see n. 4 above). As it

I spaced the marks from 60 to 120 at intervals of 1=4 inch (roughly 0.625 cm), which leftof a unit, I could only use approximations on an instrument of this size.

A. Barker / Studies in History and Philosophy of Science 40 (2009) 344–351 351

pivoted to change the instrument’s pitch-range, one has to takecare to move both the components at the same time, since theyare fastened together only by the pivot at one end. (One might finda way of fixing them together temporarily at the other end too,while the instrument is in use, but this is unnecessary.)

Despite my unmistakably amateurish workmanship and theunsophisticated materials I used, the device in this form servesthe purpose for which Ptolemy designed it fairly adequately.There are various ways in which it might be improved, some ofwhich, I imagine, would only be noticed by an instrument-makingspecialist (and such a person would certainly want to use moreappropriate kinds of wood). But a few of them are obvious evento me. I have already pointed out that a larger instrument wouldgive better sonority and would make it possible to adjust thestrings’ positions more accurately, and that metal rods might bepreferable to wooden battens as mounts for the strings. I suspectalso that the quality and volume of the sound might be improvedif one used steel strings, rather than the nylon ones I fitted to myinstrument; but this would exacerbate a problem that already af-fects the present model, though it is not too troublesome. Eventhough the wood I used for the bridges is quite hard, the tautstrings tend to make little dents or grooves in it, especially onthe fixed bridges, and steel strings would cut into the wood moredeeply, so making it more difficult to move the strings freelyalong the bridges’ surfaces. Clearly the bridges should be madeof an even harder material, perhaps metal or porcelain, or the‘artificial ivory’ which instrument-makers sometimes use for cer-tain purposes; but I have not yet tested these possibilities. If read-ers can suggest other improvements I shall be glad to hear aboutthem.

People who were present at my demonstration at the WhippleMuseum will testify that the version I have made may be crude,but that it nevertheless shows that Ptolemy’s instrument can bebuilt and made to work in practice. He seems to have been contentto leave the choice of materials to the instrument-maker himself,no doubt wisely, since an experienced craftsman is likely to be abetter judge of such things than a mere mathematical scientist.Apart from that, only a few details needed to be invented hypo-thetically (most importantly the manner in which the stringsshould be mounted). Ptolemy’s own treatise provides almost allthe necessary instructions, and none of his prescriptions is eitherirrelevant or obstructive to the instrument’s success. If he everglances at my working model from his home in the Elysian Fields,with the telescopic vision of a disembodied spirit, he may justifi-ably think it a poor specimen; but I am confident that he will rec-ognise it as in all essentials an instrument of the sort that hedesigned.

References

Barker, A. (2000). Scientific method in Ptolemy’s Harmonics. Cambridge: CambridgeUniversity Press.

Barker, A. (2007). The science of harmonics in Classical Greece. Cambridge: CambridgeUniversity Press.

Creese, D. E. (2009). The monochord in Greek harmonic science. Cambridge:Cambridge University Press, in press.

Ptolemy. (1930). Die Harmonielehre des Klaudios Ptolemaios (I. Düring, Ed.).Göteborgs Högskolas Arsskrift, 36. Göteborg: Elanders boktr. aktiebolag.

Ptolemy. (1989). Harmonics. In A. Barker (Ed.), Greek musical writings, Vol. 2.Harmonic and acoustic theory (pp. 270–391). Cambridge: Cambridge UniversityPress.

Theon Smyrnaeus. (1878). Expositio rerum mathematicarum ad legendum Platonemutilium (E. Hiller, Ed.). Leipzig: Teubner.