Abstract

French double-manual harpsichords are in wide use today, especially those based upon the widely admired extant originals by Pascal Taskin. In this paper the scaling and original stringing of Taskin’s instruments—and a few other French Baroque instruments—is scrutinised in different ways. A first conclusion is that there appears be no evidence for a recent grouping of Taskin’s instruments in “low pitch” and “high pitch”. A second conclusion is that there is no direct correlation between pitch and Taskin’s changes in case sizes and scalings: he partially compensated for these changes by suitably modifying the stringing lists. The third conclusion is that the presumed inconsistency of Taskin’s stringing lists appears also to be a myth: with reasonable pitch assumptions, his tension curves show great similarity, being also similar to those calculated for other 18th century French instruments. The fourth conclusion, highly relevant for the harpsichord maker, is provided in the form of a unified, reliable and accurate method for the calculation of stringing lists for modern instruments based on 18th century French originals. Hints to select the Taskin original to be copied are also provided.

NOTE: This article (Parts 1, 2 and 3 combined) is now available as a PDF file from Academia.edu.

Contents

1. Introduction
2. Taskin's scalings and sizes
3. Taskin's scalings and implied pitches
4. Gauges, tensions and formulae
5. Studying scalings and stringing lists
Part2 - Analysis of scalings and stringings

1. INTRODUCTION

When we try to reproduce an extant ancient harpsichord and its sound as closely as possible, two matters are crucial: scaling and stringing. They are essential in tone production, yet, until quite recently, they did not get the accurate study they deserve. The relationship between scaling and pitch is quite obvious, and equally obvious is that “Since historical instrument designs were developed to enhance and modify the output of historical wire to produce the sound desired historically, the historical sound cannot be achieved by using a different wire in an historical design” (Irvin 2010b). Moreover, there is no doubt that “The way an instrument is strung is clearly one of the most important factors contributing to the sound it produces. … the composition of the traditional stringing materials, red brass, yellow brass, and iron, … the diameter of the strings …”. (O’Brien 2010). The goal of this paper is to shed new light on these complex matters. Bibliographical references are grouped at the end.

This study owes a lot to the learned harpsichord maker Paul Y. Irvin from the U.S.A., who kindly helped the author in his initial studies on the matter, provided reading materials and very useful advice. Please note that Mr. Irvin has not checked this paper: the author is the sole responsible for its contents.

We will use here the common Helmholtz pitch notation CC, C, c, c’, c”, c’”.
Equivalences in another common system are C1=CC, C=C, c=c, c1=c’, c2=c”, c3=c’”.
To designate accidentals, rather than denoting all of them as sharps (C#, D#, F#, G# and A#), the author prefers to follow the use that was common in the “harpsichord and unequal temperaments” era: C#, Eb, F#, G# and Bb.
"Boalch2" = Boalch 2nd ed. (1974). "Boalch3" = Boalch 3rd ed. (1995).

Fig. 1 - The Ruckers-ravalé Taskin 1780 harpsichord of the Paris Conservatoire collection, upon which the author often practised from 1969 to 1971.
Commonly named "the Ruckers-Taskin in Paris", this was originally a transposing Ruckers dated 1646: it was aligned in the 18th century,
then ravalé by Blanchet in 1756, and finally had a peau de buffle and genouillère added by Taskin in 1780.
[Boalch2 = Ruckers 115. Boalch3 = RUCKERS, A. 1646(2)]
(Photo taken by the author with permission in 1970)

2. TASKIN’S SIZES AND SCALINGS

In a thorough 25-page report (Brauchli 2000) on the Taskin 1782 in Portugal [Boalch2 = Taskin 3b, Boalch3 = Ruckers(A), A(A). 1636(A)(1)], two paragraphs are of particularly relevance here. The first one, entitled "CASE DIMENSIONS", states that "The length of Taskin's double-manual harpsichords varies between 2210mm and 2360mm. The instruments can be divided into two groups: six instruments measuring 2286mm to 2350mm, and three smaller ones from 2210mm to 2250cm (sic). The 1782 Taskin, with its length of 2317mm, belongs to the larger group." (Brauchli 2000 p. 37).

Further below in Brauchli's report there is a related statement in a paragraph entitled "STRINGING": "Taskin used two scalings a semi-tone apart. The Boston, Milan, Victoria & Albert and Colares instruments are short (c2 = c.344mm) and the Edinburgh, Yale, Brussels and Paris instruments are long (c2 = c.357)." (ibid. p. 44). Unfortunately Brauchli does not clarify to which of the three choirs this c2 refers, but with the lengths he reported, it could only be the long/lower/back 8' choir. Summarising his two statements, Brauchli divides Taskin harpsichords in two distinctive groups based on their total length and scaling, and presumably also pitch. Let us call this the "two-group hypothesis".

Indeed, the use of two different pitches has been reported for Flemish mid-17th-century harpsichord making, based on the "abundant evidence, both from extant instruments and from contemporary documents" (Koster 1994, p. 50). However, when scrutinising scaling details for the Taskin 1781, Koster (ibid. p. 52) carefully avoided any implication that the two-pitch system applied also to 18th-century French instruments. As for Brauchli, his grouping includes an important contradiction: in his first statement quoted above, he included the Colares 1782 in the "large" group on account of case length, yet in the second statement he included the same instrument in the "short" group on account of scaling. Furthermore, the present author has collated Taskin's scalings and stringing lists long before reading Brauchli's paper, and found no evidence for anything like a clear-cut separation in two groups.

In order to elucidate the above conundrums, let us scrutinise the involved case lengths and scalings. A first observation is that comparisons of scalings based on c"(c2) only, as practised customarily in modern times, yield very coarse results. It is not difficult to find pairs of ancient harpsichords with scalings almost identical for c"(c2) and very different instead in the midrange and bass. Any such comparison should, at the very least, include also the vibrating string lengths of c and c'. Since case lengths are also relevant, for the 8' we should also consider the scaling for the bass, typically represented by the notes FF and C. The Table below shows these data for most of the surviving harpsichords by Taskin. The dimensions are in millimetres.

Table 1 - Taskin’s harpsichords - Length and scalings in millimetres – Ordered chronologically (1)

Notes to the table above:

(1) This table lists the Taskin harpsichords mentioned in Brauchli's paper, except the ones in Yale and Paris. For these two instruments, as well as a further two extant Taskin harpsichords (one also in Paris and the other in Bucharest), the data needed for this study has not been available

(2) Although the case length reported by Brauchli (ibid. p. 37) is 2317mm, Boalch (1974 and 1995) has instead 2337mm. Either way, this instrument can never be considered “short”. When Brauchli (ibid. p. 38) grouped it with those having c2=c.344, he mistakenly used the scaling he reported on p. 36, where the 345 clearly belongs to the column labelled “Upper manual, 8’ stop”, i.e. the short 8’, not the long one. (The author has fully investigated the possibility that the label in the column was wrong: this is certainly not the case, because the values would be too low, yielding two inconsistencies, one with the total case length, and the other with the 4’ scaling and its implied pitch.) In order to compare this instrument with the others, we need the scaling for the long 8’, not included in Brauchli’s paper: this information is not publicly available, and for this study the author has extrapolated it from the short 8’ scaling, taking into account the geometry of the 8’ bridge. The error of the results is estimated to be not larger than one millimetre per string.

(3) From the few string lengths in the Victoria & Albert Museum Catalogue (Russell 1968 p. 47), and accurate measurements of a soundboard photo (Russell 1973 Plate 50), the author has extrapolated the vibrating lengths of all the long 8’ strings with negligible error. Anyway, we do not need the utmost accuracy in this particular instrument, for reasons commented further below. This harpsichord is also the only instrument in Table 1 that has no 4' choir.

(4) The date in the signature was wrongly reported as 1780 in Boalch2 and also elsewhere. The correct date is reported in Boalch3 and has been further verified (Bonza 1997). Scalings are very similar to the 1781 instrument, but larger in the extreme bass of the 8’ and bass of the 4’.

Many conclusions can be drawn from Table 1 above. It is immediately apparent that there is no absolute consistency of case length with scaling: no two instruments are identical, and there are actually pairs of instruments (all of them double-manual, the only single being the 1786) such that one has a slightly longer case while the other has a slightly longer scaling. However, these particular discrepancies are minimal in size. The list above can easily be sorted by "scaling and then case length", as shown in the following Table 2.

Table 2 - Taskin’s harpsichords - Length and scalings - Ordered by scaling and length

The data in Table 2 resolves both conundrums mentioned above. A first conclusion is that the "two-group hypothesis" is clearly untenable. Also, no two instruments are identical in scaling and size. Furthermore, if we look at the full string length data and try to group them, scalings in the 2nd and 3rd octave seem to show three groups, and up to five groups are needed if we also consider bass and treble scalings. Finally, Brauchli's above-mentioned contradiction about the Colares Taskin 1782 is also resolved: this instrument is neither "short" nor "long", but "medium" instead.

Let us list interesting peculiarities of some of Taskin's instruments, which will be relevant further below:

• The Goermans 1763-Taskin 1784, first instrument in Table 2, has a peculiar bridge shape in the bass. The author in 2011 visited this instrument in the Russell Collection in Edinburgh, where the curator Dr. Darryl Martin kindly showed the evidence that, in spite of Taskin's visible gap-surgery to add a peu-de-buffle stop, bridges and nuts are where Goermans (then aged 61) put them in 1763. The large scaling was surely meant for a pitch lower than the other instruments in the Table.
• The Boston 1781 has a peculiar bridge shape as well, with a shorter scaling, surely meant for a much higher pitch.
• The Milan 1788 has a scaling very similar to the 1781—except in the low bass—suggesting a similar pitch.
• The London 1786 instrument "could possibly have been made for Louise-Honorine, Duchesse de Choiseul (died 1801), who was herself unusually small. She was an accomplished player and had small furniture specially made for her." (Victoria & Albert Museum, undated webpage). It is indeed an almost Lilliputian instrument, with only one keyboard and two 8' choirs, and it cannot be meaningfully grouped with any other French Baroque harpsichord (find more details in Russell 1968). Nevertheless, it has been included in this study because it provides useful information.
  

3. TASKIN’S SCALINGS AND IMPLIED PITCHES

The "New grouping" in Table 2 above is by size and scaling. Due to the reciprocal relationship between string length and pitch, it would seem intuitive—and explicitly stated by Brauchli in the STRINGING paragraph quoted above—that grouping by size automatically implies grouping by pitch, "long" implying "low pitch" and "short" implying "high pitch". Let us call this the "scaling vs pitch assumption": were it true, it should be possible to match Taskin's scalings with pitches in use in 18th century France.

For the purpose of this analysis we can initially assume the "short" 1781 instrument to have had a pitch not unlike the one we know Taskin used three years later: his already-mentioned tuning fork of 1784 at A=409 Hz. It is very important to bear in mind that this initial hypothesis is for comparative analysis only, i.e. it simply provides a convenient "origin of coordinates": it is irrelevant for absolute pitches, as will be clear further below.

Other things being equal, string length and pitch are inversely proportional: therefore, if the "scaling vs pitch assumption" holds, we can rewrite Table 2 in terms of pitch for the tuning fork A. We first fill the "1781 Boston" row with repeated values following the above assumption A=409. For the other rows, for every length L in Table 3, we look for the length B in the same column and in the "1781 Boston" row, and we compute the new value P = 409 x B / L .

Table 3 - Taskin’s harpsichords - Implied Pitches
(assuming the Boston 1781 Taskin to be at A=409 Hz)

The "New Grouping" found in Table 2 still holds true. However, something else is now seriously wrong, even disregarding the late and diminutive Taskin 1786 in the Victoria & Albert Museum. First, the pitches are remarkably inconsistent between the different ranges and choirs within the same instrument (for example, was the Brussels' Taskin 1774 pitched at very low A=363 or at reasonable A=393 or even 396?). Even more importantly, for the "scaling vs pitch assumption" to be true, we have to accept that the three medium-sized Taskin harpsichords, dated 1769-1782, were all pitched at around A=370: this is much too low compared with the known historical pitches for France at the time, which hovered around A=400.

We could try to resolve this conundrum by assuming A=400 as their average pitch, but this would then imply that the Boston 1781 was built for approximately A=442 (and other instruments in Table 3 also at unrealistically high pitches). This is much higher than any other known French 18th pitch (the highest is A=423 and comes from Lille, not Paris). Besides, at such a pitch many strings would have their stress (see section 4 below) increased, dangerously approaching their breaking point.

The conclusion is that, even though Table 3 superficially shows "many similar instruments and a few deviations", these deviations, especially for the long (1784) and the short (1781) instruments, are too large between instruments—and inconsistent within instruments—to be justified solely in terms of pitch. From our analysis above, we can only conclude that Taskin after 1780 made some full-size instruments shorter, of which the 1781 and the 1788 are the extant examples. However, their reduction in size was too significant to be explained only by higher pitches, taking into account known pitches in use at the time in France. In other words, the values listed in Table 3 are contradictory, proving that the scaling vs pitch assumption is untenable.

We have therefore to search for explanations other than pitch. Possibly Taskin made some instruments shorter so that they would be easier to move around. The main drawback of slightly shorter cases and—mainly bass—shorter scaling is a minor loss of sound quality in the bass range. In these harpsichords this only becomes noticeable below BB, and perhaps it is not a coincidence that these notes were very seldom played in these instruments, because in the post-Baroque era they were mostly used for accompaniment. As a solo keyboard the fortepiano was fast taking over as the fashionable instrument, and indeed the one to which Taskin was devoting his main efforts in the 1780's.

We will see below that a confirmation of the above findings—and a much deeper insight—is provided by a more involved analysis which includes the stringing schedules. If higher pitches cannot explain Taskin's shorter scalings, the latter should be compensated for by a heavier stringing, as otherwise the shorter instruments would have their strings with lower mass and significantly lower tension, yielding a lower sound volume and a relatively poorer "buzzy" sound. (Let us call this the "scaling vs stringing compensating hypothesis").

Thankfully, most of the extant Taskin instruments have the maker's stringing schedule inscribed in the instrument itself. We will try below to assign, to each one of these instruments, a tuning pitch that is within historical limits for the time, and see which common traits we observe in the related tension curves.

4. GAUGES, TENSIONS AND FORMULAE

When studying French Baroque stringing schedules, an important hurdle is that they are stated in gauge numbers, while modern work needs them converted into wire "sizes", i.e. diameters in thousands of inches, or else their equivalent in hundredths of millimetres. This conversion is far from straightforward, because ancient strings show some variation, and the very few extant strings and lists are incomplete and inconsistent. Not surprisingly, every decade or so a scholar publishes a new study and a different list. The author has scrutinised a recent thorough analysis of the extant sources and reconstruction of the Nuremberg wire used by 18th c. French makers (Wraight 2000). Wraight's "Nuremberg Synthesis" is a very important advance in our understanding of the old gauges. The author believes that his system can be further refined when applied to French harpsichord wire: see the full details in Appendix 1 below. The author's improved conversion list has been used for the calculations throughout this study.

The other important matter is the calculation of the tensions. There is some confusion in the musical literature (although the matter is absolutely clear in the scientific acoustics field), because the term "string tension" is sometimes used for two very different concepts, which we will name here using the unambiguous words "stress" and "pull", defined further below. The calculation formulae have been around for centuries, and are not difficult to find in the Internet. However, often they are not stated in the most convenient form for the calculation of string schedules in keyboard instruments, which is as follows.
First the stress T, which is the internal tension of the material in the string, and is expressed in Kilograms per square millimetre of cross-section (Kg/mm²), is calculated thus:

                    T = 4D(FL)²/1000G ,      which simplifies as:                       (1) T= D(FL)²/2450

Then the pull P, which is the load of the string from the instrument's frame, and is expressed in simple weight units or Kilograms (Kg), is calculated thus:

                    P = D(FLM)²/1000G ,      simpler as a function of T:           (2) P = 506.71 TW²

The above formulae use the following data:

                    D = Density of the string alloy in Kilograms per Litre (Kg/Lt)
                    F = Frequency of the note in Hertz (Hz)
                    L = Length of the vibrating string in metres (mt)
                    G = Acceleration of Gravity at sea level, average value 9.80, a constant that is integrated into the final constants in the formulae (1) and (2)
                    M = Wire size, i.e. string diameter, in millimetres, which in the final formula (2) shows instead in inches (see W below)
                    W = Wire size, i.e. string diameter, in inches (in.)                      W = M/25.4

5. STUDYING SCALINGS AND STRINGING LISTS

We do not know which method the old masters used to calculate their stringing schedules their instruments. Most likely, they would start by the schedule of a similar instrument, and by successive trial and error they would find improved selections of alloy and size for some of the strings. In the case of French Baroque makers like Taskin, small but significant variations are found in his extant large doubles, both in their scalings and in their schedules, which he usually marked in the wrestplank. Comparisons show prima facie that the latter were matched to the former. However, when looking at historical schedules in detail, one finds details that have puzzled some modern writers.

The author's opinion is that, perhaps, there is nothing to be puzzled about. It is apparent that the ancient makers decided string alloys and diameters by ear, not by calculation, and experiments show that a harpsichord string only sounds decidedly wrong when either the wire size or the alloy is relatively far away from the ideal requirement. Makers and tuners know very well that in any instrument, for some strings, a change of one gauge up (or down) can be almost inaudible. This implies that ancient empirical schedules should not be expected to be very accurate or consistent. Furthermore, ancient makers lived in a harpsichord-making tradition, following designs, procedures and materials optimised generation after generation: their results were acoustically so good (as their extant instruments can attest), that a few less-than-optimal strings in an instrument were not significant.

This situation is not advantageous for us trying to make today well-sounding good-quality replicas. There are many things we cannot replicate accurately from the past: indeed, in some respects we are much off the mark for lack of the appropriate materials, such as seasoned soundboard wood. In order to compensate for these shortcomings, we strive for optimality in variables that are under our control, such as scaling and stringing. However, we cannot simply string our instrument from an ancient stringing schedule that has internal or external (i.e. relative to others) inconsistencies, and which furthermore is unlikely to fit our instrument, whose scaling inevitably—even if slightly—differs from the original. To arrive at a stringing schedule that will sound optimally in a replica harpsichord, we need first to understand what the French 18th c. harpsichord makers were aiming at—even implicitly—when selecting their string alloys and sizes.

As said before, we intend to study a subgroup of instruments for which we know the historical stringing schedule, assigning to each one a tuning pitch that is within historical limits and also makes tension curves similar among the different instruments. We will do this below in section 6, but before It is important to clarify the results we expect from our project if successful:

1. It will confirm that the decreasing size of Taskin's harpsichords was not solely related to pitch, and that the loss in tension was avoided by suitable changes in the stringing list.

2. It will disprove the generally held notion that Taskin's—and indeed French 18th century—stringing schedules are highly inconsistent. (Let us call this the "schedule inconsistency hypothesis").

3. It will provide a solid guideline for modern makers trying to find an optimal stringing schedule for replicas of Taskin's harpsichords and other French 18th century models.

4. It will also provide useful suggestions for the selection of the ideal Taskin model to copy, based on the desired tuning pitch.

The Tables above have included seven instruments by Taskin. Luckily, for all of them—except the one dated 1769—we know the stringing schedule, because Taskin inscribed it in the wrestplank. However, six instruments with their schedules are a small sample, and to have more instruments is really desirable. A first instrument we will add to our study is the Taskin 1769 in Edinburgh: even if no stringing evidence is extant, we can use our new reconstruction of the stringing schedule, included here in Appendix 2.

It could be argued that, since our reconstruction is correlated to other schedules, the evidence it adds to them is statistically weak. This is true, and in a "weighted average" of schedules, this one should be assigned a relatively low weight. However, its inclusion is important because this instrument, with its scaling that is different from others by Taskin, might show pull curves that differ in shape from other instruments: if this does not happen, its inclusion is a "passed statistical test", adding weight to our conclusions.

Let us finally add a further two harpsichords, not by Taskin and earlier in date: the Ruckers 1612 (Boalch2 =31) ravalé before mid-18th century and now in Paris, and the Hemsch 1736 (Boalch2 = 4) now in Boston. For them we will use a general ancient mid-18th century stringing list (Corrette 1753). Having more and earlier instruments makes it significantly more difficult to "unify by tension": therefore, if we succeed, our argument is significantly stronger. Find below the complete list of nine instruments—and their schedules—for the analysis that follows.

Table 4 - A chronological list of French 18^th-century double-manual harpsichords with known scalings and stringing schedules
[“Boalch2”: number in Boalch 1974, 2^nd edition. “Boalch3”: number in Boalch 1995, 3r^d edition]

(*) Scalings available only for the F’s and C’s. The full list of string lengths has been reconstructed for this research using an algorithm for “parallel curve fitting” with similar contemporary or Taskin instruments: the error has been estimated to be smaller than 0.3%.

Three further ancient French schedules are available, but it was decided not to include them in the present study:

• Blanchet 1733, B2, in Thoiry. Hubert Bédard is reported to have measured remaining original strings from which a full schedule and tensions were computed (Bakeman 1974 p. 110). We have reverse-calculated the string lengths, which are mostly consistent with other French harpsichords of the time, except however for the extreme bass, where the length would be ridiculously long compared with any extant French instrument. In addition, the tensions suggest a very low pitch, not higher than A=383. These contradictions arise because many stages in the calculations are inevitably error prone, yielding scalings and stringings that are much too inconsistent and unreliable to be included in any serious study.

• Dumont 1697, B1, ravalé Taskin 1789, in Paris. A schedule has been reported (Bakeman 1974 p. 111) but no ancient source for it is mentioned in the paper: it may simply be a modern stringing by Bédard.

• Dumont 1699 (this instrument is not included in any Boalch ed.). Hubbard (1965 pp. 208-209) transcribes a stringing list, different from the one reported by Bakeman for the DUmont 1697. With these uncertainties for the two Dumonts, as well as their early dates, it has been decided not to include these lists in our study.

Fig. 2 - Engraving from Michel Corrette's continuo tutor (Corrette 1753).

Before embarking in an involved analysis, it is good to perform a first check for the "compensating hypothesis" (end of section 3): having seen that the "short" Taskin 1781 has scalings too small for its pitch, the hypothesis implies that the stringing should be heavier to compensate. Sure enough, if we collate its schedule with the ones we know for the "medium" sized instruments, most strings in the Taskin 1781 have the same or larger diameter. Having successfully passed this test, we can proceed with a full study of the schedules.

For each one of the harpsichords in Table 4, the first thing to do is to calculate, using computer spreadsheets, the "pulls" of all the strings in the long 8' and 4' choirs. Then we have to find, for the above instruments, suitable tuning fork pitches: ideally they should be historically likely and possibly also make the pull curves consistent: one tries different values for the tuning fork pitches until a consistent result is reached, or else an inconsistency revealed.

INHARMONICITY. One tries to find common traits in ancient pull curves, but inharmonicity is a matter that was also audible to ancient makers (Louchet 2009 Chapter 5). This is because no real-life string exhibits the ideal theoretical behaviour: strings that are too thick relative to their length are the worst offenders. First comes the top treble of the 4' choir: here the inharmonicity is very significant, but due to the high fundamental pitches, the out-of-tune overtones are not a serious problem when either tuning or playing. Then comes the extreme treble of the 8', where the same amount of inharmonicity would be much more audible: but inharmonicity is much smaller here, thus also irrelevant in practice. In the extreme bass of the 8', however, inharmonicity of a few single strings is easily heard. It is also proved by inconsistencies when tuning by octaves: it is not unusual to find problems when tuning an octave with two 8' choirs, whereby either one of the unisons or one of the octaves is not pure, and the tuner has to find a compromise tuning. In historical instruments with shorter scaling, where one would expect larger wire sizes to avoid "buzzy" low pulls, there is actually evidence of a smaller size having been used in the extreme bass. Obviously the maker found that these thinner strings, with their lower inharmonicity, actually sounded better. We will find below more consequences of inharmonicity in the extreme bass.

Please read the continuation of this study in Part2 - Analysis of scalings and stringings

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