NON-PYTHAGOREAN PROPORTIONS AND THE DIAPASÓN IN THE DESIGN OF IBERIAN HARPSICHORD SCALES

One of the tenets of modern organology is that the strings of keyboard instruments have generally been designed according to the theory that length is in versely proportional to pitch, resulting in a progression of string lengths kno wn as a Pythagorean Scale. Recent research has begun to focus on surviving Iberian instruments, heretofore largely neglected. While few have survived, it would appear that a signifi cant number of their builders had a tendency to employ non-Pythagorean scales. One recent study suggests that these instruments represent a curious collecti ve adherence to archaic methods used in the design of Medieval instruments. The present article of fers an alternative viewpoint, showing how such scales could easily ha ve been constructed using a traditional organ builder’s design tool, called a Diapasón. The geometric logic of the Diapasón is briefl y explained as well as the v arious progressions which can result from its use. The scales of se veral Iberian instruments are then examined in light of these alternative methods, arriving at the conclusion that their makers, rather than exhibiting an archaic conservatism, actually formed part of growing 18 century movement towards the use of extended non-Pythagorean scales as are now ubiquitously found in the modern piano.

The Iberian harpsichord has recently begun to receive an ever increasing amount of attention from historical organologists after an initial period of relati ve neglect in the early decades of the modern harpsichord revival.The newest addition to this rapidly gro wing body of literature is John K oster's examination of the w ork of a handful of instruments representing the Valladolid school, in which he uses these e xamples as a springboard for e xamining the Iberian harpsichord' s hypothetical lineage within the larger context of European harpsichord making.K oster illustrates how the Iberian mak ers drew various elements of style and design from other schools of making, most notably Flemish and Italian1 .One aspect, however, is singled out by Koster as being unique: the design of the lengths of the instruments' strings, or the "scale" as it is usually called.
Koster's reasoning is based on the hypothesis that so-called "Pythagorean" scaling, in which the lengths of the strings follow a reduction which is in versely proportional to the fundamental frequencies they sound, is the basis of all non-Iberian harpsichord making (with a fe w occasional exceptions found mostly in rectangular virginals).A signifi cant number of the surviving Valladolid harpsichords, however, have markedly non-Pythagorean scales.Since the a wareness of Pythagorean proportions was omnipresent in European culture, Koster argues, these Iberian makers must have had some extraordinary motivation for their "collecti ve indifference to the niceties of scaling as practiced almost e verywhere else"2 .His hypothesis is that these Iberian mak ers may ha ve been emplo ying archaic design practices hark ening back to Medieval examples, using large circular arcs to defi ne the shape of the instrument's bridge, and points to the f act that such methods were still commonly emplo yed in the 17th and 18th centuries in architectural design and the construction of buildings.He concludes his article with a recreation of ho w one of these builders, Joseph Bueno, might have drawn his bridge shape in just such a manner, using two large arcs with centers well outside the boundary of the instrument, the fi rst, with a radius of about 100 cm, defi ning the upper half of the scale, and the second, with a radius of about 270 cm, defi ning the tenor portion until the sharp turn towards the spine, called the "bass hook".
The present article presents a different view, demonstrating that while Pythagorean scaling does indeed seem to be the ideal for the Italian school, non-Pythagorean scales are not as rare in northern European instrument making as K oster posits.When vie wed against this broader vision of what constitutes "normality", it becomes evident that, rather than representing a curious hark ening back to "archaic" practices from the 15th century, these Valladolid makers were actually scaling their strings in a manner similar to more adv anced builders working elsewhere in Europe, employing design aspects which would eventually become adopted by all modern piano mak ers in the second half of the 19th century.In addition, it is demonstrated ho w a methodology commonly emplo yed by or gan mak ers perfectly explains the peculiar shape of all b ut one of the non-Pythagorean scales found in the Valladolid instruments.The number of truly puzzling scales is reduced to those of only tw o instruments, and a 3 The instruments are the 1599 Celestini (Fig. 1a) in the Museum für K unst und Gewerbe, Hamburg (2000, p. 509), and the 1772 Shudi & Broadw ood (Fig. 1b) in the Museum of Fine Arts, Boston (1977, p. 57).The string lengths are tak en from BEURMAN, Andreas: Historisches Tasteninstrumente.München, Prestel, 2000, and KOSTER, John: Keyboard Musical Instruments in the Museum of Fine Arts.Boston, Museum of Fine Arts, 1994.4 The data for the instruments presented in this article come from a v ariety of sources; in some cases, the length of e very note was known, but in others, only the lengths of a small number of notes were reported in the literature, and the rest of the string lengths were extrapolated by spline curve fi tting methods.In the latter case, the reported lengths are indicated by lar ger outlines for the individual plot points on both the absolute and proportional traces.Whenever such indications are absent, the analysis was done with measured lengths of every note.
5 Proportional deviation curves in units of semitones are, in my opinion, the best method of judging the design aspects of any scale, for the y make subtle changes in scaling v ector immediately obvious to the e ye.This method of analyzing instrument scales was developed by me in the early 1990' s.By contrast, the often encountered logarithmic plot of string lengths, despite its proportional basis, suffers from a number of drawbacks: fi rst, the scale of the display is far too small, making it all but impossible observe the overall shape or any localized alterations within the scale; secondly , the data plots all run at highly oblique ang les to the vertical axis, meaning that it is exceedingly diffi cult for the eye to correctly judge the magnitude of the v ertical distance between any two plots ; and fi nally, the basic unit of visual measure (log length) has no immediate relationship with rele vant aspect of the physical object under study, in that while one may be able to see a certain distance between tw o traces, the graphical representation alone provides no way to judge its signifi cance.In essence, log plots make all but the most markedly deviant scales appear to "more or less" follow Pythagorean proportions, and their common use may well be one of the reasons why modern or ganologists have generally missed the existence of many non-Pythagorean scaled instruments.
as it represents a functional de viation of only about one-third semitone6 .Furthermore, as can be seen by observing the general shape of the bridge in the tenor re gion, because of the highly oblique angle of the bridge relative to the strings, the slightest de viation to the left or right of the ideal position can cause rather lar ge absolute de viations in string lengths.This fact, when combined with the reduced proportional signifi cance, means that we should accept aberrations which are much lar ger in absolute terms in the tenor regions than in the treble when evaluating the credibility of any hypothetical design ideal.In both of the instruments, there are minor de viations from the ideal curv e, possibly caused by original workshop error or distortion of the instrument o ver the centuries.Nonetheless, the errors are v ery small both in absolute and proportional terms, and the original intent is therefore quite clear .The scale of the Celestini is Pythagorean until the sudden deviation of the "bass hook" for the last seven notes.While such drastic "foreshortening" of the lo wer notes is not uncommon, Italian instruments often continue to follow a strict Pythagorean progression much further into the bass.
This instrument, for e xample, would have a bottom string about two meters long if "correctly" scaled; some Italian instruments which have larger compasses can arri ve to string lengths of almost 230 cm.F or comparison, a dashed trace indicates the Pythagorean curv e of the Schudi/Broadw ood iron-strung portion, sho wing how an ironstrung instrument, if made with Pythagorean proportions deep into the bass, would be even longer yet.
The Shudi/Broadwood is strung in iron in the treble and therefore has a generally longer design curve.The scale is therefore Pythagorean only above c 1 , and from there down, the notes are gradually foreshortened, the bridge more or less follo wing a straight line.When the strings reach the general length of a brass Pythagorean scale (indicated by a dashed line), the stringing material changes from iron to brass, which is then used for the remainder of the lower notes.
Koster presents a number of e xamples which adhere to this sort of scheme from the Flemish, German, and English schools, and briefl y outlines the supposed design methodology behind all such instruments 7 .The assumption is that the method used by Ruck ers was practiced universally; a length was determined for the top note of the instrument, usually c 3 in the early instruments, and w as then doubled twice to obtain the lengths of the notes c 2 and c 1 (with some occasional minor v ariation at c 1 ).These three octaves of the note c were the "key notes" which the maker used both conceptually to design the scale and practically to actually mark the position of the bridge on the soundboard.If desired, other key notes may have been calculated for greater accuracy during bridge placement, using simple Pythagorean proportions; the lengths for the octa ves of f, for example, can be derived by multiplying any of the c lengths by 1,5 and then halving and doubling for all other octa ves.Koster concludes his examination of "typical" northern scaling by summarizing that "the uni versal principle incorporated in all these scaling recipes was that the upper half of the compass should be Pythagorean, with strings doubling in length for each lower octave from c 3 down to c 1 .Even when the actual c 1 string was made a little shorter, the scaling remained close to the theoretical ideal down to about f1" 8 .While all of K oster's examples are 16th and 17th century instruments, I ha ve chosen an 18th century harpsichord to illustrate Type 2. The Iberian harpsichords under examination are also 18th century.This fact is not insignifi cant, for starting in the closing decades of the 1600's, it becomes less easy to fi nd examples of scales in northern European instruments which adhere to strict Pythagorean proportions from around middle c 1 all the w ay to the top note.Instead, we often fi nd signifi cant deviations in the upper re gions, signifi cant in that the y are too great and the original intent is too evident in order to e xplain them a way either as w orkshop error or subsequent distortion.The clear majority of these deviations involve a region near the top of the instrument which has been purposely "stretched", that is, the lengths have been augmented beyond the supposed Pythagorean design lengths.In the mildest of cases, this stretch is only about a semitone, but many instruments demonstrate trebles with are augmented by as much as two or three semitones.Figure 2 shows a noteworthy example, the Taskin instrument from the Russell Collection in Edinb urgh 9 .Here, lik e the Shudi/Broadw ood, the Pythagorean section clearly begins at c 1 , but in this case, it spans only about an octa ve and a fi fth.From f 2 , the last octa ve has clearly been augmented until f 3 is tw o semitones longer than the main part of the curve.This deviation might be considered a fl uke were it not for the existence of at least two other instruments by Taskin with exactly the same deviation in the high treble 10 .The scales of such instruments still might be considered "essentially Pythagorean" in their upper half in that the mak er could ha ve designed them by using simple Pythagorean proportions for the notes c 1 , f 1 , and c 2 , plus some method of adding a little amount for c 3 and/or f 3 .However, a number of instruments simply cannot be considered as ha ving anything like Pythagorean proportions as the basis for the treble half of their scales.Two examples which are exceptionally instructive can be found in one and the same instrument, the 1777 combination harpsichord/fortepiano or Vis-a-vis built by Stein, now in Verona11 .This large rectangular instrument has k eyboards at both ends, resembling a harpsichord and a piano joined along their bentsides.This instrument gives us a unique opportunity to compare the scaling of harpsichords and pianos at a time when both were being made by eliminating all possible incongruities which could result from dif ferences of time, place, mak er and pitch le vel; the only difference here is that the piano is more heavily strung than the harpsichord12 .Figure 3 shows the scales of both halv es of the instrument with corresponding Pythagorean curv es generated from their respective c1 notes.Three points are immediately obvious: (1) the difference between the overall lengths is almost exactly two semitones; (2) in both cases foreshortening clearly begins at the note c 1 ; (3) the shapes of the curv es for the treble half of the instrument(s) are almost identical, follo wing a markedly non-Pythagorean re gular logarithmic progression with an octa ve ratio of about 1,9:1 until leveling-off at about c 3 .The fact that these tw o scales are quite consistent in their strict logarithmic nature while being precisely tw o semitones apart in o verall length implies that Stein w as calculating his string lengths using some method that gave him irrational numbers; that is, he was not working in simple units of any local measuring system.
Finally, to sho w that Stein w as by no means alone in using truly non-Pythagorean augmented scales, Figure 4 sho ws two other instruments with similar curv es13 .Despite being the product of tw o different national schools and separated by almost 50 years, these tw o scales are remarkably similar .Due to the slopes of these curves, it is diffi cult to say with absolute certainty where the foreshortened tenor ends and the treble scale logic be gins, though both scales seem to mak e a change of v ector around tenor c.Both scales follo w decidedly non-Pythagorean progressions for most of the treble.As with Stein, the near straight-line v ectors for a signifi cant portion of the de viation curve indicates that the progression are logarithmic; the dashed lines indicate a re gular progression with an octa ve proportion of 1,82:1.In a similar manner to Stein, both scales appear to fi nally level-off at about f 2 , more or less follo wing a "normal" Pythagorean progression for the highest octa ve only.Once again, the near perfection of these slopes over a range of two octaves would indicate that both builders were using some means of calculation which allo wed them to operate freely within a uni verse of irrational numbers, rather than being sla ves to the often assumed methodology of calculating scales using only whole units or simple fractions of a local measuring unit.
Before continuing with a detailed analysis of the scales of the Iberian instruments, ha ving demonstrated that northern European builders were not unfamiliar with the concept of non-Pythagorean scales, it would be benefi cial to briefl y consider the possible motivations behind the use of augmented scales.When an instrument mak er sets out to design the lengths for the series of notes comprising a musical scale of se veral octaves, he must operate within the bounds of tw o extremes: strings which are too long, producing a tension which w ould cause the strings to break before the y could be tuned at the desired pitch, and strings which are too short, meaning that the tension le vel is far below that at which the strings would break.While the maximum length is easy to determine both theoretically and practically, defi ning a tension level which is "too low" is extremely diffi cult.In fact, there is no hard and fast rule which can be adopted, neither theoretically nor one which can be observ ed in real instruments.While the lowest notes of almost all instruments are markedly too short, this study is limited primarily to the upper regions of the scale and the transitional zone to the tenor foreshortening (when present).
Within these regions, a tension level which is too low can cause a number of undesirable phenomena, including a high degree of inharmonicity, problematic interaction with the excitement mechanism (be it hammer or quill), poor tuning stability with v ariations in temperature and humidity , diffi culty in tuning, and a noticeable drop in pitch as the tone decays from the initial moment of attack14 .
It is a generally accepted tenet of modern organology, therefore, that string lengths for the treble portions of an instrument were chosen in order to maintain the highest practical tension level possible at the design pitch, meaning that the strings were tensioned to a le vel almost at which they would break, with only a small safety margin 15 .The issue is often clouded by a common confusion between tension and stress (or load), two related though separate aspects of scale design.Tension is the absolute amount of force applied to the string through extension, commonly measured in kilograms, while stress is the amount of force relative to the string's cross sectional area, often expressed in kilograms per millimeter squared (kg/mm 2 ) 16 .Breaking stress or rupture load is the stress level at which a string will break, and it is this aspect -not tension per se -which the b uilder must consciously control in the design of his scale, for if he fails to keep the lengths within certain limits, he will not be able to tune the instrument to the desired pitch level before the strings break 17 .
The generally assumption among organologists has been that breaking stress is material-specifi c and does not vary with diameter.Heavier strings will require more tension to sound the same pitch at a given length, but the increase in tension will be e xactly compensated by the increase in the string' s crosssectional area.Putting it in practical terms, for any given string length, it is assumed that all diameters of the same material will break at the same pitch.Conversely, for any given pitch, all diameters should require exactly the same length to maintain any desired safety margin.Taking it one step further, then, since length and pitch are al ways exactly inversely proportional on a Pythagorean scale, the logical conclusion is that all diameters of a gi ven material located an ywhere on a Pythagorean scale are all equally close to breaking.Conversely, one can calculate that particular Pythagorean scale which is of suffi cient length to achieve the desired safety margin for all diameters, and this scale can be seen as a sort of standardized maximum for the material, which can then be represented by the length of but one note: c 2 .
As widely held as it is, the basic assumption which underlies this entire chain of logic is not true.As wire is reduced to smaller diameters, it work hardens, exactly as when it is formed with a hammer and anvil.Therefore, breaking stress is not the same for all diameters of the same material, nor there is one single Pythagorean curve which represents the "maximum safe scale length" for an y type of wire 18 .Smaller gauges are stronger relati ve to their area, and therefore will al ways be under-stressed on a Pythagorean scale, because the working stress level, which is constant, becomes ever lower compared 16 Common as they are in the literature, these units are not scientifi cally correct since they both use the (kilo)gram as a unit of force, while it is properly a unit of mass.Often an attempt is made to resolve this problem by using the unit of kilograms force (kgf), though this is no real solution because it is gravity acting upon the specifi ed amount of mass which is the operati ve force, and the force of gravity varies depending upon the distance from the center of the earth.Proper scientifi c usage is to represent tension in Newtons and stress or load in Pascals.17 Stephen Birkett, who has been the fi rst in modern times to attempt to reproduce music wire with the same metallur gical composition as ancient wire, reports that his research indicates the the elastic limit of the wire w as just as much a limiting factor as rupture load (personal communication).The elastic limit is the point at which the wire be gins to permanently deform, which can be located anywhere along the stress/strain curv e from several semitones up to a small fraction of a semitone belo w the rupture load.Strings with a lo w elastic limit will not break immediately , but rather at each tuning the y will be found to be slightly fl at; after having been raised again numerous times, the y will eventually break at the supposedly safe pitch le vel.As important at this aspect of the behavior of wire may eventually prove to be, the issue is far too complex to be dealt with here.See BIRKETT, Stephen: "Historical iron music wire and its practical modern replication as a viable commercial product", in Proceedings of the 2006 harmoniques International Congress, (in preparation).
18 As Goodway and Odell put it, "the choice of the gauge [diameter] of wire... is also choice of the strength of the wire."The italicized emphasis is that of the original authors.See G OODWAY, Martha, and ODELL, Jay Scott: The Metallurgy of 17th-and  18th-century Music Wire.Stuyvesant, Pendragon, 1987, p. 63. Malcolm Rose iron wire Type A, for example, reaches a pitch/length strength parity with his brass wire at a diameter of about 0,55 mm, and from there on, progressi vely larger diameters of the ir on wire are actually weaker than the corresponding diameters in brass, meaning the iron w ould requiring a shorter scale.This turns the assumptions of modern organology completely upside down.
to the rising load-bearing capability .The most highly stressed string will always be found at or just below the lowest note located within the Pythagorean zone, for it is precisely there that we fi nd the fatal combination of the lar gest (and therefore the weak est) diameter with the longest o verall scale length 19 .
In fact, if a builder wants to keep his wire consistently critically-stressed, as is commonly assumed, he must use a non-Pythagorean progression in which the o verall scale length is continuously augmented by an amount which is e xactly proportional to the increased strength introduced by each change to a smaller diameter.When we realize this, the issue is turned completely around.Actually, the question is not, "Why did some b uilders use non-Pythagorean scales?", b ut rather, "Why didn't all builders take advantage of tensile strength pick-up by intentionally emplo ying augmented (i.e non-Pythagorean) scales?"The answer depends some what on stringing material; brass wire e xhibits a much lo wer rate of increasing strength with decreasing diameter , so little that an y length adv antage to be gained is hardly worth the ef fort of emplo ying a mar ginally-augmented scale.With the much higher pick-up rate of iron wire, ho wever, the answer is most lik ely that many builders simply didn't care enough to go to the extra effort of testing all the various diameters of the wire and calculating the corresponding non-Pythagorean proportions required for a consistently-stressed scale.F or these b uilders, most notably the Flemish makers, it was far easier to keep to the simplistic mentality of ancient Pythagorean proportions, allowing them the further practical adv antage of using standard rulers mark ed in normal units of measure to position their bridges.The fact that the strings in the upper re gisters were understressed by two or three semitones was either unknown to them (though that is diffi cult to imagine) or of no consequence to them; otherwise, they would have designed their instruments differently.
Other builders, however, such as those whose instruments ha ve been presented here abo ve, decided to take advantage of this e xtra strength, augmenting the length in order to gi ve their treble notes a cleaner, clearer, more singing tone.In that sense, the y were not only emplo ying a more sophisticated approach to the design of their scales, indicati ve of a more complete understanding and manipulation of the "ra w material" which is iron wire, b ut the y were also more forw ard-looking, predicting the practice found in man y late 18th and early 19th century pianos and ubiquitously from about 1870 onward.Modern pianos are generally built with augmented non-Pythagorean scales with octave ratios of about 1,9:1.As we shall soon see, the majority of the survi ving Valladolid harpsichords indicate that their builders as well were among those looking forward toward the future rather than back towards the past, as proposed by Koster.
The Iberian harpsichords e xamined here consist of fi ve out of K oster's six central instruments plus one other which was only briefl y mentioned by him.Our e xamination will begin with the last of these, an instrument whose mak er is unknown, though he is thought to ha ve been from the pro vince of Salamanca20 .Koster provides a plan view of the instrument and mentions that its scale is "suitable for brass strings throughout the compass", though no detailed description of the shape of the scale is given.Figure 5 demonstrates that this instrument is of the Type 1 scale design, having not only a short overall length (c 2 = 274 mm) "suitable for brass", but also a strict Pythagorean scale all the way down to almost tenor c, where a brief portion of foreshortening suddenly and rapidly leads directly to the bass "hook".Thus Iberian mak ers were perfectly capable of making instruments with strict Pythagorean proportions when they wanted to.
The other six instruments pro vide a v ariety of scale shapes, all of which are more or less non-Pythagorean in their treble scales, some of them mark edly so.Therefore, the y are indeed unlik e either the generic "Italian" model (with short or long tail) or K oster's proposed generic "northern European" design.Ho wever, prepared as we no w are with a broader vision of northern b uilding styles, a ree xamination will sho w that we can e xclude four of these instruments from ha ving scales which might be characterized as demonstrating "a collecti ve indifference to the niceties of scaling as practiced almost everywhere else."In so doing, an interesting detail will emer ge which may well prove useful for future studies re garding instruments of all kinds.Two instruments, ho wever, will remain stubbornly puzzling, one of which will be excluded from this study because of the uncertain nature of the data provided21 .A possible explanation for the last instrument's markedly aberrant scale shape will be provided here, though by no means do I intend this hypothetical solution to be an ything other than that: possible guidelines for further lines of inquiry.
The anonymous instrument attributed to Fernandez Santos in the Museu de la Música de Barcelona (MDMB1495) is the easiest to e xplain22 .Its scale, sho wn in Figure 6, is remarkably similar to the last two northern instruments illustrated abo ve; from tenor c upw ards, the scale follo ws a v ery strict logarithmic non-Pythagorean progression with an octa ve ratio of 1,88:1.The o verall scale lengths among the three instruments are consistent within a range of less than one semitone, indicating that all three could well have been designed for the same general pitch (a 1 ≈ 415 Hz).The only difference is that the Santos (attr.)slope is a bit less steep than that of Gräbner and Tibaut, which is precisely why the maker was able to continue following this augmented scale right up to the top note without le veling-off at around g 2 , as in the other two.In all cases, the total augmentation found in the range of the smallest gauge (usually the highest 10 notes or so) is about three semitones relative to the length of tenor c.We can assume that all of these scales were designed to maintain a consistent stress le vel over the entire range of the instrument from tenor c upw ards, a supposition which is supported by an y number of believable solutions for an iron stringing schedule 23 .
23 The v ariety of dif ferent solutions possible according to dif ferent hypothetical diameter distrib ution schemes (i.e."stringing schedules") and tensile strength pick-up rates is v ast, and I hesitate to cast doubt upon the general conclusion by presenting any one example which will inevitably contradict some minor detail of one or another author' s interpretation of these variables.The dubious reader is invited to explore the matter and perform a variety of complete tension and stress calculations for these instruments employing values taken from a variety of sources, as I ha ve done, after which it will become ob vious that any solution based upon a generally accepted range of variables will be in general agreement with the analysis presented here.The next two harpsichords are transposed instruments and therefore ha ve markedly different scale lengths from all the others, the fi rst sounding a quint pitch and the second at the octa ve 24 .Despite this difference in length, however, they both have remarkably similar scale shapes.As can be seen in fi gures 7a and 7b, tenor foreshortening clearly begins just below middle c1.The upper regions follow a non-Pythagorean curve which results in a total augmentation of approximately two semitones at the top note.The slight reduction in the total amount of augmentation is of little signifi cance, as it would only mean that the b uilder wanted to have a slighter greater safety mar gin in the highest notes.What is peculiar is the f act that these progression are not re gular, as in the instruments seen abo ve.Any regular logarithmic progression will produce a straight upw ardly-sloping deviation trace, b ut these instruments both e xhibit a curv e which sags do wnward in the middle of the o verall rise at the note 24 The instruments are Anonymous (probably Valladolid), Obradoiro Instrumentos Musicais, Lugo (K oster App.1/4) and Anonymous (probably Valladolid), museo Nacional de Antropología, Madrid (K oster App.1/5).String lengths used for the extrapolation are those reported by Koster.c 2 .It's possible that this peculiar shape is due to original w orkshop error or later distortion, b ut the remarkable similarity of the two scales would imply that it is an artifact of original intent.As we shall soon see, this particular shape is a direct product of the method by which these scales were most lik ely devised.This methodology, although commonly emplo yed in ancient instrument making, has no w been largely forgotten; curiously, despite a signifi cant number of documentary references, it has also received almost no attention from modern organology.
As e xplained pre viously, the supposition that Pythagorean proportions produce the most natural model for the string lengths is not true if the instrument is strung in iron wire and the b uilder wishes to maintain a reasonably-consistent level of stress.Under these conditions, he has no choice b ut to consciously design a non-Pythagorean scale in such a manner that it is augmented in the highest notes by an amount proportional to the increase in strength between the thick est iron string in the tenor and thinnest string in the  treble.While this may seem a daunting task for an ancient mak er with no recourse to modern metallur gical testing equipment, it is actually quite simple to achieve.The methodology was precisely described by the Viennese piano maker Jacob Bleyer in 1811 25 .Using a monochord, the builder fi rst determines the "best" lengths for two notes at the top and bottom of the re gion which is to follow some sort of a regular curve, i.e. the region above the foreshortened tenor; in Ble yer's case, his k ey notes were tenor f and f 4 .Bleyer did not state his criteria for defi ning "best", b ut it is safe to assume he meant the longest length with a reasonable safety mar gin.Once these tw o lengths are kno wn, the lengths of the interv ening notes were interpolated by constructing a "geometric series" between the tw o extremes.The method by which such progressions were calculated in ancient times was by employing a triangular geometric calculator (hence the term "geometric progression"), known among instrument makers as a diapason.
The use of the diapason to calculate the dimensions of pipes among 18th and 19th century French organ builders is well kno wn, as diapasons are illustrated and described in both l'Encyclopedié and Dom Bedos' famous exhaustive treatise of the subject 26 , as well as a number of lesser -known works.A diapason w as also illustrated to defi ne the circumferences of a rank of or gan pipes in Arnaut von Zwolle's often-cited 15th century manuscript, though he called it a fi gura fi mbrie 27 .The diapason was also known and used in Spain in the design of both or gans and stringed instruments, as is pro ven by the following quotes 28 : "Diapasón [...] entre los or ganeros, es una declinación formada en un plano, que demuestra todas las longitudes, y latitudes de las fístulas, o caños del órgano, y por cuyas medidas se van cortando: su fi gura es la de dos líneas, que empezando con alguna anchura, caminan rectamente a unirse a un punto, aunque no llegan a él" 29 .
- 29 "[...] among or gan makers, it is a reduction in the form of a plan, which sho ws all the longitudes and latitudes of the pipes, or tubs of the organ and according to the measures of which the y are cut.Its fi gure is that of two lines, which begin with a certain width [between them], running straight towards a point where they join, although they don't arrive at it".
30 "To obtain the best quality of instruments, it is v ery important that one poses a proportioned ruler , which the makers call diapasón in the vulgar.And not only should this be observed in the strings, but also in the tubs of the organs, in which, they must have the proper proportion, the length and the width have to be proportioned.This is how the organ builders always do it".
In both modern Spanish and Catalan, the w ord has also come to mean the fi ngerboard of instruments such as the violin, the guitar or the lute, probably in reference to the proportional distances between the positions where one places the fi ngers or frets, a division which is exactly the same as that of the base line of an or gan maker's diapason.Diapason also means a standardized reference frequenc y and/or the tuning fork from which it can be taken, a defi nition shared by Italian, French, Portuguese, Catalan and Spanish 31 .However, of all these Romance languages, Spanish is the only one with an offi ciallysanctioned dictionary in which the ancient defi nition can still be found today: "Regla en que están determinadas las medidas con venientes, en la cual se ordena con debida proporción el diapasón de los instrumentos, y es la dirección para cortar los cañones de los ór ganos, las cuerdas de los clavicordios, etc." 32 .This linguistic tenacity may well be an indication of the application of the diapasón to the problems of musical instrument design among Spanish makers which was even more common than in other European traditions.
Such a widespread and enduring popularity among musical instrument makers of all kinds would be well-deserved, for a diapason is a powerful and fl exible tool, easy to construct, eminently practical in the w orkshop environment, and especially applicable to musical instruments in that it f acilitates calculations involving irrational numbers.With a diapason, there is no need whatsoe ver to defi ne the diameters or lengths of pipes or strings in any local rational unit of measure, such as pulgadas, palmos or varas; dimensions can simply be transferred to it or taken from it by copying them directly from or onto the work at hand, or to a dedicated marking stick 33 .More important, it releases the b uilder from the intellectual bonds of the simplistic Pythagorean proportions observ able in the design of man y Italian and Flemish instruments, allowing him to construct a wide variety of logarithmic progressions as well as comple x irregular progressions, feats which might otherwise seem be yond the capabilities of "simple" artisans with no formal education in mathematics.Actually, considering the large number of dimensions needed for a rank of pipes and the f act that they progress by exceedingly fi ne degrees, it w ould literally be impossible to w ork in simple fractions of an y normal unit of measure.Or gan builders probably would have found the discrete unit methodology of the Flemish makers to be naively simplistic, being as the y were completely accustomed to mo ving about within a v ast uni verse of numerically undefi ned dimensions which had been determined geometrically.
The precise nature of the progression obtained from a diapason is not al ways the same, ho wever.Depending on how the diagram is constructed and used, the results can either be a strictly logarithmic Pythagorean series (steps approximating 12 √2 with lengths halving at the octa ve), a logarithmic non-Pythagorean augmented progression (halving at interv als larger than the octa ve), or a wide v ariety of irregular series with a progressively-increasing augmentation.Therefore, before considering the hypothesis that the scale of any particular extant musical instrument -be it an or gan, a harpsichord, clavichord, piano, or guitar -may ha ve been designed using a diapason, it is absolutely essential to understand the theory behind these geometric calculators as well as the shapes of curves they can produce.The simplest form of diapason is sho wn Figure 8.A horizontal line is dra wn at an y arbitrary length con venient for the medium, be it paper , parchment, or a w ooden board 34 .This line is then divided into tw o equal parts, and one part is again equally subdi vided, the process being continued until there are as man y progressively-halved segments as the number of octa ves over which the scale is to be calculated.The longest of these segments is then partitioned into 12 proportional subsegments representing the 12 notes of a chromatic scale, after which the rest of the se gments are partitioned in a similar manner simply by halving the lengths along the line from the right end 35 .After the line is 34 The 1594 inventory and assessment of the valuables of the organ builder Andrés Gómez in Toledo contains the entry "dos tablas de diapasones", literally "two boards of diapasons".The fact that they were included is an indication of their status as one of the essential tools in the organ builder's workshop, and therefore a valuable asset.See SAURA BUIL, Joaquín: op.cit., p. 171.35 The methods used for di viding the fi rst se gment into 12 proportional steps were not necessarily al ways precisely logarithmic.Simple Pythagorean proportions were used both by Arnaut and Bedos; l'Encyclopedié offers both a "Just Intonation" solution, in volving the ratios of pure fi fths, fourths, thirds and sixths, and an equally-proportional di vision (i.e."Equal Temperament"), although the v alues given for the latter are not correct.In an y case, the slight irre gularities from step to st ep produced by the different methods are insignifi cant for the purpose of scaling.partitioned, the v ertical leg of the triangle is added on the left.This fi rst ordinate represents the fi rst dimension of the series to be scaled, either in full size or at some simple scaled reduction (di vided by any convenient whole number).Finally , the triangle is completed by adding the hypotenuse, and the remaining ordinates are determined by dra wing lines upw ards to it from e very mark along the base line.The heights of the ordinates will follo w whatever proportional series was used to partition the base line, in this case a (quasi)-Pythagorean series, in which the ratio between an y two adjacent elements is the 12 √2.Once the basic diagram is completed, an y number of scales can be calculated simply by marking a different height for the fi rst ordinate and striking a new hypotenuse.In each and every case, the ordinates defi ned thereby will follow the same proportional reduction rate of 12 √2.
To the untrained e ye, all diapasons may appear to be alik e, but nothing could be further from the truth.While such a Pythagorean diapason illustrates the basic principle of geometric calculation, it had almost no application in the or gan building trade, since almost nothing in an or gan follows a strict Pythagorean scale36 .Organ builders discovered long ago that man y dimensions of or gan pipes, primarily those relating to cross-sectional area and various mouth/reed dimensions, could not be scaled according to Pythagorean proportions, for if all dimensions of the pipes were simply halv ed or doubled at the octaves, the longer pipes w ould be too lar ge and the shorter pipes too thin in order to function well and maintain a good overall balance of volume over the entire gamut37 .Therefore they used scales in which these dimensions reduced more slo wly than by halving at e very 12th step, that is, scales in which the lengths were gradually "stretched" or augmented compared to Pythagorean lengths, and their diapasons were especially constructed to produce these non-Pythagorean scales.
Figure 9 shows an example of a regular logarithmic augmented-scale diapason of the type which became common among or gan b uilders in the 19th century 38 .Here the halving distance is greater than the normal octa ve interval of a Pythagorean 12 √2 progression, which is achie ved by partitioning half of the base line into a series of proportional steps whose number is greater than 12, in this case 14½.Since the ratio between each successi ve step is 14½ √2, starting from the ordinate at an y note, its half value can be found by counting up 14½ steps (black arro ws have been added to mark the initial successive halving points along the base line).Organ builders have developed a common terminology which refl ects this underlying logic; this scale w ould be referred to as "halving on the 15½ th " (note, or line on the diapason, inclusive the fi rst).Any n √2 can used as long as n > 12. Exactly as with a 12 √2 diapason, the diffi cult task of partitioning the base line need only be done once, and other scales with the same proportional reduction rate b ut of dif fering overall lengths can be dra wn simply by adding more hypotenuses.As long as they all converge with the base line at its right-hand extreme, the ratios between any two ordinates defi ned by an y given hypotenuse will al ways be the same ratio used in the initial partition of the base line.Although there is no documentary e vidence for the construction of this type of scaling triangle before the 19th century , the f act that such re gular non-Pythagorean progressions were used by harpsichord mak ers is pro ven by the instruments e xamined above which have an upwardly-sloping deviation trace which follows a straight line 39 .
Before the 19th century, however, the most common type of or gan builder's diapason differed in both structure and result from such re gular designs.They are dra wn in a manner similar to a re gular diapason, be ginning by successi ve halving of the base line and further partitioning according to a 39 Regular logarithmic non-Pythagorean scales can also be calculated by another type of or gan builder's diapason, one based upon a spline curve.For a complete explanation of this process, see Pol et t i, Paul: "Beyond Pythagoras...", op.cit.(quasi) 12 √2 series.The critical difference comes at the moment of drawing the hypotenuse; in this case, two dimensions are used to defi ne two ordinates, those of the fi rst last pipe of the entire series, no matter ho w many intervening notes there are between them, and then the hypotenuse is dra wn by connecting their upper e xtremes 40 .However, since there is no agreement between the halving of the abscissae and the halving of the ordinates, the geometric unity which is the v ery essence of a normal diapason is destroyed.In reference to this critical dif ference introduced by the manner in which the slope of the hypotenuse is defi ned, I call this type of scaling diagram an "αω (alpha-omega) diapason".Arnaut's method is identical to that of Bedos with one e xception: instead of defi ning the last pipe in the series, he defi nes the last ordinate at the right end of the diagram, which represents the dimension of a pipe which does not e xist, being an infi nite number of octaves above the fi rst pipe.Nonetheless, the shape of his scale is identical to that of any normal diapason and can be accurately calculated by taking the ordinates of the fi rst and last pipes from his fi gura fi mbrie.
41 While it is technically incorrect to refer to the upper sloping side of such a trapezoid as a "hypotenuse", I shall continu e to use this term both out of convenience and as a reminder that the basic logic of the αω diapason remains grounded in triangular geometry, even when this logic is purposely subverted.
As this plate clearly illustrates, one can dra w dif ferent hypotenuses on an y pre viouslyconstructed αω diapason base, just as with the re gular diapasons, b ut in contrast, there is no guarantee that there will be an y correlation between the progressions defi ned by the v arious trapezoids.Each height/slope combination will interact in a dif ferent way with the logarithmic 12 √2 partition of the base line, creating an infi nite number of irre gular progressions.Short of actually measuring each and e very ordinate of each indi vidual αω diapason, it might seem an impossible task to predict the progression each defi nes, let alone arri ve at a "General Theory" applicable to all possible αω diapasons.Luckily, understanding the geometry of αω diapasons and predicting the nature of the scales the y defi ne with mathematical e xactitude is f ar simpler than one might assume.
As mentioned above, the initial steps of partitioning the base line are identical for both the αω and the regular Pythagorean version.It has also been demonstrated that a strict Pythagorean triangle can result with an infi nite variety of hypotenuse slopes as long as the base and hypotenuse both terminate in a single convergence point.Logically, then, for the hypotenuse of any irregular αω trapezoid, there also exists a theoretical Pythagorean triangle which would have a base line located at whatever height is required for convergence between its end and the end of the hypotenuse.This means that inside every αω trapezoid, there hides a regular Pythagorean triangle; fi nding the triangle is the k ey to understanding the αω design.To fi nd this hidden triangle, a ne w base line is simply e xtended from the point where the hypotenuse meets the shorter of the tw o vertical sides of the trapezoid.This virtual base line restores the unity between the halving of the abscissae and the ordinates, as sho wn in Figure 11.Thus the αω trapezoid can be viewed as consisting of tw o components: a re gular Pythagorean triangle standing on top of a long rectangle.The series of dimensions it defi nes consists of the regular Pythagorean series ( 12 √2 ) of the triangle augmented by a constant, k, which is equal to the height of the rectangle.In reference to this compound derivation, I will refer to such a series as a "P+k scale".
The exact nature of each series depends on two design characteristics: the slope of the hypotenuse and the height of the rectangle relati ve to the height of the shortest ordinate defi ned by the triangle.In general, though, k is usually relati vely small compared to the Pythagorean ordinates, b ut as the regular triangle reduces in height, k becomes ever more signifi cant.If the progression is carried out far enough, the roles reverse, and the Pythagorean ordinates become much less signifi cant compared to k, essentially eliminating any increment whatsoever between successive values.Therefore the scale of an αω diapason transforms itself from a quasi-logarithmic progression of near -identical proportions to a quasi-static succession of near-identical absolute values 42 .Considering this inherently metamorphic structure, the nomenclature " αω" acquires an additional signifi cance, for these scales represent a continuos connection between the two opposing paradigms of stringed musical instrument design: the constant tension/varied length model, the basis of the harp and all harp-like instruments, including the clavichord, harpsichord and piano, and the constant length/varied tension model, the basis of the open strings of the lute, guitar, and violin families.
P+k scales have a distinct appearance when they are graphed as deviation from a Pythagorean series, as sho wn in Figures 12a and 12b .Figure 12a represents an analysis of three octa ves of the P+k series generated by the top line of the Bedos diapason illustrated in Figure 10, assuming tenor c as the starting note.The x axis represents a normal regular Pythagorean progression, and the bold 42 I suspect that many ancient instrument makers did not understood the mathematical logic made evident by this analysis.For them, it was simply a practical method of creating a non-Pythagorean series of diminishing dimensions.Ho wever, even without understanding the theoretical basis for the de vice, I have no doubt that ancient mak ers were well-versed in its use and certain ly would have had a vast empirical knowledge about the sort of scales produced by a variety of height and slope combinations.

Figure 12a
Figure 12b curved trace sho ws the P k curv e. F or comparison, v arious other progression are also illustrated: the regular log series required to arrive at the same amount of augmentation at c 3 (12,8 semitones); three common 19th century "rationalized" pipe scale progressions, halving on the 16th, 17th and 18th steps (light gray lines); the maximum historical pipe scale, halving on the 24th; and fi nally, a constant series using the length of c 3 for all notes.The angles of approach where the P+ k curve meets the Pythagorean scale and the constant length scale are similar, indicating the transformative nature of the curve, smoothly bridging the gap between the two extremes, and giving the P+k curve its unique shape: a downwardly-sagging curve.Figure 12b shows the rest of the series generated by the various hypotenuses of the same diapason.While they all vary more or less from one another , they all share the same sagging de viation curv e.Therefore, an y time an augmented scale with a sagging curve is detected, the chances are high that the series was generated using an αω diapason.This is exactly the type of curv e which can found in the treble re gions of fi ve of the se ven Iberian harpsichords under discussion.
Armed with this kno wledge about the appearance of a scale generated by an αω diapason, the sagging curves of the two anonymous transposing instruments are no longer puzzling.Table 1 shows that the lengths of the reported key notes of these scales agree quite well with the corresponding v alues of a P+k series.Thus it w ould seem highly probable that these mak ers were in f act using or gan b uilders' methodology to produce mildly augmented treble portions of their scales.In so doing, the y were simply following a trend in instrument design which w as becoming gradually more common throughout the 18th century , fi nally culminating in the common practice among late 19th century piano mak ers.Bleyer's 1811 description is echoed by Julius Blüthner in his Lehrbuch des Pianofortebaus, originally published in 1872, in which he instructs the no vice that the scale of the upper octa ves should not follow strict Pythagorean proportions, b ut rather should be augmented by as much as the "quality" of the wire allo ws" 43 .Slightly dif ferent methods are gi ven in dif ferent editions, the earliest being a strict logarithmic progression, while the 1886 edition gi ves a solution which is a P+ k progression.This seeming indifference to the precise manner by which the scale is augmented is also seen in the Valladolid harpsichords, with Santos (attr.)Barcelona being strict logarithmic and the tw o transposed instruments being P+k.
The remaining Iberian instruments do indeed mo ve further a way from an y set of hypothetical norms which might be derived by examining instruments of other traditions, and understanding them presents some very interesting challenges to the organologist.The scales of the Santos 44 and the Bueno instruments are illustrated in Figures 13a and 13b .They are some what similar in that the upper tw o octaves are augmented non-Pythagorean progressions which appear to ha ve been designed from middle c 1 upwards using an αω diapason, as shown in Table 2.The two upper octaves of the Bueno 45 exhibit such an incredible de gree of agreement with a P+ k scale that it w ould seem unlik ely to ha ve been designed by an y other method.The octave below, however, appears to ha ve been scaled do wnward from middle c 1 by simple Pythagorean doubling.Both instruments have a puzzling dip in scale length between c 1 and c, as well as a slight rise just above the bass hook; these aberrations from the supposed 43 BLÜTHNER, Julius: Lehrbuch des Pianofortebaues in seiner Geschichte, Theorie und Technik .Leipzig, 1872/1886.Blüthner's comments are highly informati ve both as to the moti vation and the process of emplo ying augmented scales, and are therefore worthy of reproduction in full here.He introduces the topic in both editions with the same text: "Nur in den oberen Oktaven bestimmt man die Saitenlängen annähernd so, als wären alle Saiten gleich dick und gleich gespannt, d. h.man reducirt die Saitenlänge beim Aufsteigen um eine Okta ve auf die Hälfte,.Wie schon erwähnt, v erfährt man aber in der Regel nicht nach diesem Principe...sucht man nämlich im Allgemeinen die Mensur im Diskant zu vergrössern, so weit die Beschaffenheit der Saiten dies erlaubt, um denselben mehr Elasticität und den Tönen mehr Stärke und Gesang zu geben".s(Only in the upper octaves does one determine the string lengths as if all strings were equally thick and equally tensioned, that is, one reduces the string length by half with every octave ascending.As already mentioned, one as a rule does not follow this principle...but one generally attempts to augment the treble scale as much as allo wed by the quality of the wire, in order to give the same more elasticity, and to make the tone stronger and more singing).
(The exact determination of the lengths v aries quite a bit.As an example, the values given by Sievers will be tak en here.For the a of the tuning fork ( a 1 ), he takes a length of 370 mm for the vibrating part of the string, using the gauge 17.The a 2 one octave higher should therefore have a string length of only 185 mm, b ut one gives it 194; likewise, a 3 gets 102 mm instead of 97 and a 4 54 instead of only 51 mm.).
(For the a of the tuning fork ( a 1 ), one takes a length of 40 cm for the vibrating part of the string, a 2 gets a length of 21 cm [40/2 + 1], a 3 gets a length of 11 cm [40/4 + 1], and a 4 is taken at a length of 6 cm [40/8 + 1].Be yond that, if the instrument is made short or long, this has no infl uence of the length of the strings of these four octa ves.The string lengths of the bass octa ves are determined by the length of the instrument).
44 Andrés Fernandez Santos, Valladolid, 1728.Collection of Fernanda Giulini, Briosco.45 Joseph Bueno, Valladolid, 1712.Fundación Joaquin Díaz, Urueña.design logic are probably simply due to the spline curve 46 which naturally results from connecting the key note lengths.46 The bridge on a harpsichord follo ws the mathematical shape called a spline curv e precisely because it is an e xample of the object from which the term is deri ved, a "spline": a w ooden batten or other thin semi-rigid object which has been bent t o follow a series of points.Any spline will always adapt somewhat gradually to any change of vector precisely because its capability to suddenly alter course is limited.While the design methodology of these instrument seems to be identical, the results, however, are drastically different.The scale of the Santos, much like the trio of Tibaut/Gräbner/ Santos (attr.)Barcelona instruments seen abo ve, is proportionally longer than tenor c by about three semitones at the top note.As has been sho wn, this amount of augmentation is perfectly understandable when the builder has chosen to profit from the increased strength of the smaller sizes of iron wire.The Bueno, ho wever, has an astounding amount of more than six semitones augmentation, which is f ar too lar ge to be attrib utable to the e xploitation of an y reasonable amount of tensile strength pick-up 47 .Whenever a scale is found which is as se verely augmented as is the case here, one possible explanation is that the instrument originally had the same number of string choirs distrib uted among a narrower range with the inclusion of sub-semitones, often called "double sharps".The additional choirs needed for the non-enharmonic notes would require a partition of the octave into more than 12 steps.As Koster points out, the range of C/E to a 2 (42 notes) was quite common in Iberian k eyboards of the 17th century.If the current 45-note range of C/E to c 3 where reduced to the earlier span, three extra choirs w ould appear.One solution might be to gi ve one e xtra accidental to each of the three octaves, most likely an ab in addition to the standard meantone g#48 .However, while doing so might help reduce the excessive augmentation of the upper two octaves, it would only make the dip between tenor c and middle c w orse.In order to reduce the e xcessive augmentation, a solution w ould have to be found which places all of the extra notes above middle c.Another reason why they might be found only in the upper octaves is that such chromatic notes would be primarily used as thirds or sixths, and never as the roots of triads.
A close e xamination of the plan photograph gi ven by K oster suggests that the 8' string for the highest note has been added later .There is not quite enough room at the top of the bridge to maintain the normal spacing between the last bridge pin and the penultimate pin, and therefore last choir spacing is too narro w at the bridge, the string running at an ob vious oblique angle to provide suffi cient space for the jacks at the gap.The hitchpin is also squeezed into the tiny space in the bentside/cheek corner, being very close to the pin for the note b .Likewise, the tuning pin also seems to be uncomfortably close to the inside of the cheek.On the 4' bridge, it appears that the lowest note is also pinned at a reduced spacing, being only about 70% of the a verage distance for the four notes immediately above it.At the 4' nut, the second string is pinned at a distance from the bass end which is more or less equal to distance between top pin and the treble end of the nut.All of this implies that the lowest 4' string was also a later addition.The instrument may have originally had the range of C/E to a 2 with tw o sub-semitones in the treble (44 notes), one each for each of the upper tw o octaves, with both choirs plucking left.At some later time, the mo ve to modifi ed meantone temperaments made the use of sub-semitones obsolete, and a lar ger compass to c 3 had also become prevalent.To convert the instrument, an extra string was added to the top of the 8' and the bottom of the 4', extra mortises were made at the bottom of both re gisters, and the 8' jacks were turned facing right.Naturally, since the spacing at the tails would be one semitone too wide where each sub-semitone was removed, a new keyboard would have to be made, remo ving any traces of the original compass in the process.I w ould be the fi rst to admit that such a suggestion should be tak en as nothing more than hypothesis until the instrument itself could be e xamined more carefully .In an y e vent, it is well known that a number of survi ving instruments ha ve indeed under gone such a transformation, so it is by no means outside the realm of possibility .In light of this f act, reconstructing the original design process using a diapason to calculate the string lengths is e xtremely instructive, for it not only demonstrates the vast difference between working in this manner and the usually assumed method of devising string lengths, but it also shows how the results can vary depending on the exact procedure chosen.
From the v ery be ginning, the use of a diapason is fundamentally dif ferent from the Douwes/ Ruckers approach, for the need to concei ve of the upper re gions of the scale as a block of multiple whole octaves is eliminated.By defi nition, the values used for an αω diapason can be separated by any number of notes.Therefore, even though the top note of the instrument is a 2 , there is absolutely no need to think of the rest of the key notes as lower octaves of the top a.While this may seem illogical to those accustomed to the Ruck ers tradition, documentary e vidence indicates that or gan builders did indeed think this way49 .Thus the hypothetical task of designing the upper notes of an instrument which ends with a 2 can just as well begin from middle c 1 .
Let us suppose that the builder has tested his wire on a monochord and found that the diameter for the highest notes breaks at a pitch two semitones higher than the thicker diameter he will use at c 1 , and therefore he wants to augment his scale at a 2 by two semitones.Furthermore, he has determined that the ideal length for c 1 which provides the desired safety margin is 28 local units, so 28 will be his α value.His fi rst step, then, is to discover the value of ω for a 2 .To do this, he marks 28 units on the fi rst vertical line (c 1 ), counts up to the 25th line (c 3 ), marks 7 units (28/4), and places a straight ruler connecting these two dots.The hypotenuse hereby defi ned gives him a Pythagorean scale, a representation of any equal absolute stress situation for all diameters.Then, starting at the line for a 2 (line 22), he counts back two notes to the left to g 2 (line 20), sets his compass to this length of this ordinate, and transfers this dimension to line 22, effectively stretching the length for the note a 2 by exactly two semitones.He now has his αω values and is ready to proceed with the rest of the scale.Exactly how he does so, however, will produce subtly different solutions.
When an or gan builder w ants to scale the widths or diameters of a rank of pipes with subsemitones, he need not include the e xtra notes in his calculation, for he can simply use the same dimension for both pipes.Of course, he may well choose to mark an extra ordinate very close to the normal accidental, b ut the dif ference between ideal dimensions for g# and ab will be e xceedingly fi ne, hardly worth the effort 50 .The harpsichord maker, however, must choose between two options; he can either ignore the fact that a diapason gives dimensions for all of the notes of the series, working like some makers in other traditions and using only one k ey note per octave to position his bridge, or he can tak e advantage of the diapason' s complete calculating capability and use more than one note per octave, increasing the precision of his bridge placement.If he choses for the former, he can forget about the extra notes and simply count up the same number of lines as normal found between his key notes (inclusive) to fi nd each length.The spline curve which the bent bridge describes will automatically a verage out the irre gular distrib ution of note lengths around the added accidental, much in the same manner that a diatonic harp neck often follo ws the shape of Pythagorean curv e even though the pitch dif ference between successi ve notes is sometimes a whole step and other times a half step, creating localized jumps abo ve and below a true Pythagorean progression in the actual string lengths.However, is he choses to use some other note near the middle of each octa ve, he must take the extra choirs for the sub-semitones into account.In this case, he w ould have count up the same number of lines as the actual choirs of strings in each octa ve, including the extra subsemitones.50 Bedos, for example, fi rst partitions the base line of his generic diapason by a series of ratios for pure fourths and fi fths for notes C through B, and then commences again from C to partition the notes he calls F , Bb, Eb.G#, and C#, using the same proportions.In strict temperament terms, this would correctly give the positions of Ab and Db.Not only did he intermix his sharps and his fl ats, but he is also using the method for determining a Pythagorean tuning in a time when the applied temperament would have undoubtedly been either ¼ comma meantone or some v ariant thereof ( temperament ordinaire).Thus it is ob vious that the fi ne nuances of proportional calculation required for temperament do not translate to the considerably coarser process of scalin g.See BEDOS DE CELLES,François: op. cit.,.Table 3 give the result of the different methods for a scale with both one and tw o sub-semitones per octa ve.Option 1 sho ws the v alues for the k ey notes c 1 , c 2 , and a 2 for the one note per octa ve approach, using 22 lines on the diapason from the fi rst to the last note (inclusive).Option 2 shows the same values for a multiple k ey note approach with one sub-semitone per octa ve, using 24 lines total, and Option 3 the same approach applied to a tw o sub-semitone plan (eb/d# and g#/ab), requiring 26 lines 51 .Note that in each case, the value for c 2 is different, becoming progressively shorter and having progressively less augmentation relevant to a Pythagorean length.Needless to say, if these instruments were later converted to a 12-tone chromatic layout, each option would result in a subtly different curve shape and length v alues for an y assumed k ey notes.The differences are small, b ut they may be just enough to throw a researcher off the track when trying to understand how the scale was devised.
This hypothetical glimpse into an ancient instrument mak ers workshop serves primarily to illustrate the complexity of attempting to de vise an explanation for those scales which in no credible w ay can be seen as being based on Pythagorean proportions.The fact that such an undertaking is comple x is no reason not to attempt it, ho wever.Before doing so, ho wever, collectively as modern researchers, we would be well-served by pondering the old adage: When the only tool you have is a hammer, every problem looks like a nail.If our analytical tools are based on the a priori assumptions of the ubiquitous adherence to whole units or simple fractions of local measuring units, the calculation of string lengths by elementary Pythagorean proportions, and the use of primary k ey notes which are always octaves of the top note, and if the method of graphical representation used to e valuate scale shapes minimizes the visual impact of aberrations from the assumed Pythagorean design curve rather than making them obvious, then the objectivity of the examination will be undermined from its inception.This very real and present bias is unfortunately further enshrined and codifi ed by the common museum documentation practice of measuring and reporting only those strings assumed to be key notes -usually octaves of the note c, sometimes with octaves of the note f -leaving the unfortunate researcher who cannot personally examine the instrument with no choice b ut to interpolate the rest of the scale.Naturally , whenever a 51 These are the two most likely place were sub-semitones would be added, for they are the most common places were the missing accidentals in meantone would be problematic.As already mentioned, ab is need to for the subdominant in c minor , and d# is needed for the dominant in E major.scale comes under examination which was created using some entirely different method, the dogmatic nature of this chain of assumptions prevents us from detecting any telltale patterns which might lead to other explanations.Under such conditions, the only remaining resort might well appear to be to reach back to archaic practices in order to justify theories based on geometric coincidences 52 .It may well be true that some Iberian harpsichord mak ers were so lacking in an y real theoretical and/or practical sophistication regarding scale design that they needed to resort to such techniques, but this would seem rather unlikely considering that man y of them were also or gan builders, and even the smallest or gan requires that one operate on a conceptual le vel which is f ar beyond that needed to fi gure out a set of string lengths for a harpsichord.In any event, before we accept the proposition that this w as the case, it would behoove us to exhaust all other possible e xplanations fi rst, especially those for which ample documentary evidence exists.
Recibido: 10/07/2009 Aceptado: 10/07/2009 52 It is common knowledge that any three points can be connected by a circular arc.It just so happens that when a scale is more or less augmented, three points tak en at the top, middle and bottom of the consistently-scaled treble portion can almos t always be connected by a circular arc which very accurately follows the shape of the bridge.Using a variety of methods, including CAD drawings and plan photographs with digital o verlays, I ha ve been able to match circular arcs with the bridge shapes of a number of 19th-century fortepianos of 5½, 6 and 6½ octa ves with a de gree of agreement every bit as good as that of the Bueno harpsichord over a range of up to 3 octa ves in the largest instruments.In fact, were it not for the kinks beneath e very strut of the cast iron frame, the upper three to four octaves of a modern Steinway would also very closely follow a circular arc.Therefore, the fact that this can be done is a demonstration of nothing except a geometrical coincidence.

9
String lengths were taken from the data sheet published on the web site of Edinb urgh University Collection of Historic Musical Instruments: http://www.music.ed.ac.uk/euchmi/uck/uckd4315.html 10 These instruments are the 1782 double in Colares, Portugal, and the 1787 double in the Museum für K unst und Gewerbe, Hamburg, Inv.Nr. 2000,532.They are described in BRAUCHLI, Bernard: "The 1782 Taskin Harpsichord, Colares, Portugal", in The Galpin Society Journal, LIII (2000), pp.25-50, and BEURMAN, Andreas: Historisches Tasteninstrumente.Op. cit.Scale analysis by the author using the data provided in the relevant source literature.

Figure 9 .
Figure 9.A rational regular non-Pythagorean diapason in which the dimensions halve 14½ steps.
Figure 10 shows a typical αω diapason, which is Figure 150 (Plate XIX) from Dom Bedos.Note that the base line is partitioned as in a re gular Pythagorean diapason, b ut e xactly as described by Terreros y Pando in 1786, none of the various hypotenuses 41 converge at the end point of the base line.If they were to be extended further until they were to cross an extension of the base line, an anarchical jumble of different convergence points would appear.The means that none of the ordinates defi ned by any one of the hypotenuses will follow a regular logarithmic progression.

Figure 11 .
Figure 11.Deconstructing the αω Diapason trapezoid into its component parts: a regular Pythagorean right triangle poised atop a rectangle.

Table 1 .
Agreement with P+k series.

Table 2 .
Agreement with P+k scales.