TASKIN HARPSICHORD SCALINGS AND STRINGINGS REVISITED
Part 2 - Analysis of Scalings and Stringings
Copyright © 2011 by Claudio Di Veroli, Bray Baroque, Bray, Ireland, March 2011
To Paul Y. Irvin
Back to Part 1
6. ANALYSIS OF 18th CENTURY FRENCH SCALINGS AND STRINGINGS
Typical common features of French 18th c. harpsichord scalings and stringing schedules are that the resulting pulls decrease in subtle ways from bass to treble. The use of the different metal alloys is also quite uniform (O'Brien 1981). On average, the transition points are as follows:
The consistency of the transitions is remarkable: none of the ancient schedules in this study deviates more than one semitone up or down from the above.
It is common in the literature to find statements such as "The Taskin 17XX and 17YY schedules are virtually identical, except for a very few strings ". Actually, however, no two extant schedules of Taskin are identical, and their similarities are as significant as their differences. Closer scrutiny provides important insights: we will show below thatin spite of assertions to the contrary found in modern publicationsthere is a strong relationship between schedules, scalings and pitches. Let us first perform a simple comparison of our seven ancient scheduleswe exclude here the reconstructed one for the Taskin 1769with no reference to either scalings or pitches. The numbers after the alloy are the original values following the French 18th c. gauge system.
Table 5 - Comparison of French 18th c. harpsichord stringing schedules.
Table 5 shows some trends and also a few inconsistencies:
The only correct way to find out the rationale behind ancient scalings, schedules and pitches is to compare their simultaneous effect: the pulls. For this purpose, a complex spreadsheet was setup, which calculates and collates the pulls and stresses of every single string for all the nine instruments listed above in Table 4. As before, it was initially assumed that Taskin had built his Boston 1781 harpsichord for the A=409 Hz pitch of his extant tuning fork of 1794. For the other instruments, we tried other historically likely pitches in order to reach similar pull curves. (Please note that, quite obviously, changing the assumed pitch simply shifts the pull curve proportionally up or down, it does not change the curves shape). Eventually it was not difficult to achieve very satisfactory and consistent results, which will be shown further below.
Let us first comment on the final tuning pitches impliedby the pull curvesfor the nine instruments in Table 4:
In most of the above instruments, if one attempts a pitch just a few Hz higher or lower, significant parts of the pull curve move away from the others. With the choices of pitches shown above, the result is an excellent overall agreement in the pull curves, as shown further below.
Note that there is no reason why the pitches we reached above for the Taskin's seven instruments studied here should resemble those in Table 3, which had been derived by size/scalings only. When one adds the stringingsas done in the study and commented abovethe final "Grouping by Pitch" becomes slightly different, as shown in the following table.
Table 6 - Taskin's harpsichords - Scalings and Grouping by Pitch
After Table 3 above we had concluded that, were pitch the only explanation for the different sizes and scalings of harpsichord, the lowest pitch should be an unlikely A=370 (and even lower). By applying instead the historical schedules and assuming similar pulls, the lowest pitch required is now the much more reasonable A=387: this is because, as calculations have shown, Taskin compensated the reduction in instrument size, not just with a higher pitch, but also with larger wire sizes. Summarising, section 5's inconsistency hypothesis has been shown to be untenable.
More importantly, although any study like this one falls short of a complete scientific proof, section 3's compensating hypothesis fits the data remarkably well. In other words, it all seems to show that, when Taskin used different scalings, he employed matching schedules, yielding pull curves remarkably similar among his different instruments. This is shown in Figs. 3 and 4 below, which include all the harpsichords in our study except for the smallish and 8'-only instrument of 1786. For uniformity's sake the curves are shown over the range FF-f'''. The latest instruments by Taskin had a range going down to EE, but we can safely ignore this low note in this study, because Taskin always strung it using the same wire as for FF.
Fig. 3 - Lower/long 8' pulls produced by 18th-century French harpsichord schedules
PULL DROP TOWARDS THE EXTREME BASS. A remarkable feature produced by all the ancient schedules is that the pull in the Red Brass decreases towards the bass. This is caused by the strings being shorter and shorter with respect to their "theoretical" lengths. In the rest of the instrument this is compensated by changing the wire size. In the extreme bass, however, very high sizes would be needed: e.g. for FF a pull of 7 Kg looks reasonable, but this requires a size .028", yielding a very audible inharmonicity. As shown by the schedule, ancient makers found that, in order to strike the best balance, Red Brass size should indeed increase towards the bass, but less so than required to keep a constant (or increasing) pull: thus they went from Gauge 2 (.0182") in D to only Gauge 00 (.0244") in FF.
PULL DROP IN THE 8' CHOIRS OVER THE TRANSITION. Another interesting feature common to most ancient schedules is that the resulting pull curve is slightly low around the transition from Yellow Brass into Iron. This is particularly noticeable in the 8' choirs (Bb to c#). The cause of this drop is that the top Yellow Brass and the bottom Iron are both strung with gauge 5 (.0133"), while for a gradually decreasing pull from bass to treble, a gauge 4 (.0148") would be needed in the transition range. Why would 18th century makers do this? The explanation is found in two facts they were surely aware of, even though not reported in any source:
DIFFERENCES IN THE 8' MIDRANGE. The pull's decrease towards the treble follows slightly different curves in the different instruments. For the Couchet-Blanchet-Taskin 1781, which exhibits relatively low midrange pulls and high treble pulls (Fig. 3), this may simply be due to its many rebuilds in the 18th century, which included moving around bridges and bridge pins (Koster 1994, p. 50), whereby the present scaling differs from the original. Anyway, for all antique instruments, under the pull of the strings, cases undergo deformation even after a few decades, let alone more than two centuries, particularly so in the midrange. They are more stable in the bass and treble, where the spine and cheek help to keep under control the shape of the case and consequently the scalings.
Fig. 4 - 4' pulls produced by 18th-century harpsichord schedules
PULL DIVERSITY IN THE BASS OF THE 4' CHOIR. Not easy to explain (especially the abnormally low values of the 1781 instrument), except that, for some reason, the sound of the bass of the 4' choir is not particularly sensitive to the wire sizes used.
PULL ABNORMALITIES IN THE HIGH TREBLE OF THE 4'. Note in Fig. 4 the peculiar drop in the range g"-b": this is caused by a reduction in wire size greater than needed by the pull progression. Why would 18th century makers do this? Because a smaller size reduces the breaking risk in strings that, due to the register gap, are longer than dictated by the Pythagorean progression: the smaller size also minimises the significant inharmonicity in these short strings. As for the rise in tension after b", this is due again to a long scaling because of the register gap. This could have been avoided by using gauge 11 (.0065"), which was available in Baroque times: however, this gauge was rarely used in harpsichords, and never in the schedules included in this study. Possibly this was because the mass of the strings would be too low to produce the required loudness, or else the wire-drawing process could not guarantee their uniform quality. As commented in Appendix 1, Corrette found that these very thin strings were only suitable for making wigs.
From the above 8' and 4' pull curves we can conclude that, for the selected instruments, with their scalings and original schedules, and assuming the hypothetical historically-likely pitches detailed above,
(a) There are no grounds for the inconsistency hypothesis of section 5.
(b) The ancient stringing schedules yield remarkably consistent pull curves, which can be used as models for present-day stringing.
Having departed from our previous ad-hoc pitches, the pull curves are now inevitably more dissimilar, but not so much. The issue is limited to the midrange and caused largely by the two worst offenders: the highest pulls are for the Ruckers c.1735 where the use of Corrette's schedule is hypothetical, while the lowest pulls are those of the Couchet-Taskin 1781 where (as detailed above) the bridges are not in their original position. If we disregard the curves of these two instruments in Fig. 3a, we can see that the hypothesis of relatively unified tension curves holds under a pessimistic assumption as well.
An obvious consequence of the above conclusions is that, if we wish to build a faithful replica of a harpsichord, before we even think about stringing schedules, it seems advisable to select an extant instrument that agrees in pitchbased on its original scaling and stringing schedulewith the pitch to which we will tune our replica. It is worth noting that, when copying an ancient instrument into a transposing instrument, most harpsichord makers do not enlarge the bass: eventually, to avoid losing one or two notes when transposing, the treble parts of bridges and case are enlarged.
Let us illustrate this matter with an example. The maker decides to build a copy of an 18th-century French harpsichord with original range FF-f''', for which the historical stringing list is known and the original pitch is estimated at about A=390Hz. The new instrument is meant to be transposing at A=392/415/440. The maker bases the A=392 pitch on the original, and does not add any further strings in either bass or treble: therefore, when transposing at A=415 the new instrument will have range FF-e''', and when transposing at A=440 the range will be FF-eb'''.
Now let us assume a common scenario with a twist: the new instrument is meant to be used mainly at A=415 and is now required to have for this pitch the full range FF-f''' (thus transposing at A=392 with range FF#-f''' and at A=440 with range FF-e'''). To accomplish this, each string has now to be tuned a good semitone above the pitch of the original. This produces two issues (obvious from what we have seen above in section 6):
1) The pull or load of each string is now higher than the original by 415/390 squared, i.e. by a very significant 13%. The resulting sound will be audibly worse than the original. The solution (widely practised in instruments built until the 1980's) is to devise a new "thinner" stringing list, whereby many strings will have one size less than the original. This will bring all the pulls down to the original value-range, yielding a good (albeit not ideal) solution.
2) The stress or internal tension of each string is ALSO higher by the same amount of 13%. All the strings are now much nearer to their breaking point than in the original instrument. Inevitably, the top strings in the brass range will break, as well as quite a few in the treble iron range. The only solution (widely practised in instruments built until the 1980's) is to use different alloys than the original. For the top of the brass there are different modern alloys that will fit the bill. Also, for evenness and sometimes also to prevent breaking risk, we may need to dispense with red brass altogether in the low bass and string it with yellow brass. As for the treble, at least the upper two octaves will need to be strung in some steel alloy, with significantly higher carbon content than any historically-based iron wire. At this point, we have significantly departed from the original instrument's sound, using non-historical wire alloys throughout most of its range.
The above conundrum has been caused by first selecting an original and then trying to adapt it to modern pitch requirements. What we should do instead, quite simply, is to start from our pitch requirements and then select, as an original, a historical instrument that matches our requirements. If we wish to base our instrument on a Taskin, we are in a particularly good situation because (as we have seen in the previous sections) extant harpsichords by Taskin cover a wide range of pitches.
Let us then list the harpsichord pitches and transpositions in common use today, matching them to the Taskin instruments we have scrutinised (excluding the very small 1786). In transposing instruments, we list the different pitches separated by slashes and, for each recommendation, we clarify which of the two or three pitches is chosen to be a replica of the original, i.e. to follow the original scaling.
Devising harpsichord schedules is a complex topic: a recent work provides a comprehensive treatment (Donahue 2007). Our final goal here, as stated in section 5 above, is to facilitate harpsichord stringing following historical pulls. The schedule of any instrument, antique or replica, is to be calculated by reference to "weighted average" and "envolvent curves" in the computer chart: they define the "pull space" of ancient schedules, as shown in Figs. 3 and 4 above . We then try to string the instrument following as much as possible the weighted average curve. We may have to depart from the latter, however, for two reasons:
For every long-8' and 4' string, the maker has to calculate both the pull and the breaking risk, using the simple formulae included in section 4 above. We only need a further information, which is provided by the wire supplier:
If the breaking limit is provided in other units, it can be converted into Kg/mm² multiplying it by a suitable coefficient: if the original unit is "Mpa" the coefficient is 0.10195, while if the original unit is "PSI" the coefficient is 0.0007031.
Finally, from B we have to calculate the breaking risk,
using the simple formula R = T / B, where T is the stress calculated as per formula (1) in section 4.
The percentage R is not expected be very accurate, because the value B that wire resellers provide comes from tests run under ideal laboratory-test conditions. In addition, actual wire strength depends not only on the alloy but also on the wire-drawing process (Louchet 2009 Chapter 4), which may vary even between batches by the same supplier. Finally, as is well known, every harpsichord string is further weakened by the torsions in the eye and around the wrestpin. All things considered, the following is a good general safety guideline:
Note also that, when initially stringing an instrument, it is best to tune it first a tone below and to bring it up to pitch after some days: under these ideal conditions the author has tested that strings will often resist up to the reported tensile strength (i.e. 100%). However, employing a risky schedule is not advisable, because the harpsichord owner needs the instrument to be available at all times for practice and performance: when a string breaks, it should always be possible to replace it and to pull it immediately up to pitch (for example during a recital), without any significant risk of breaking again.
The above considerations imply that breaking risk may force us to depart from the average pull curve for the instrument: this will happen if the instrument is to be tuned at a significantly higher pitch than the original. In such a case, the solution is to lower the pull throughout via thinner strings, which have higher tensile strength. The result is fully satisfactory, i.e. close enough to the ancient stringing, provided we manage to stayor depart minimallywithin the boundaries specified by the historical envolvent curves, which delimit the "pull space" of historical stringings.
The computer-calculated envolvent curves and average pulls for historical Taskin harpsichords are shown in the charts below, Figs. 5 and 6. As an example, they include the pulls of an optimal schedule devised by the author for an imaginary copy of the Milan Taskin 1788, to be built transposing at A=392/415/440. The curves in the charts show the pull at A=415Hz, using Vallotti as a representative of a typical 18th century temperament: this accounts for the non-smoothness of the green curve (anyway the differences between different circular temperament are not significant for stringing calculations). The breaking risk is shown by red dots and only for values above 70%. Needless to say, the instrument maker should measure the scalings in the finished copy before proceeding with the calculations. The same is valid for restringing an existing instrument.
Fig. 5 - Historical pulls for the long/back 8' choir in Taskin's harpsichords: envolvents and weighted average.
Comments on the manual selection of alloys and gauges shown in the above chart for the 8' of a modern copy:
Fig. 6 - Historical pulls for the 4' choir in Taskin's harpsichords: envolvents and weighted average.
There are no particular features to comment for the 4': there are usually no difficulties in selecting alloys and sizes to fit the historical pull envolvents, keeping breaking risk below safety limits.
The above methods have been successfully applied to the restringing of the Taskin-based Hubbard-Di Veroli harpsichord. Two further matters deserve a special treatment.
The author back in the 1980's developed a fully automated computer program for stringing list calculation: the program was successfully used by the Argentinian harpsichord maker Leopoldo Perez Robledo. A recent book (Louchet 2009) promotes a commercial service for schedule calculation: this can be both speedy and convenient, though quite obviously, for French 18th-century instruments, Louchet's system cannot possibly take into account the conclusions of the present study, which are significant as they affect the historical pull curves to be followed.
If one is familiar with advanced calculation methods, it is not difficult to set up a spreadsheet that fully automates the determination of both alloy and wire size. For each string, the procedure finds the best fit for the historical weighted average pull. If some strings are too risky, a parameter can be introduced that lowers pullsall or by groupsuntil a suitable compromise solution is reached: however, this may imbalance the pull in the brass strings, requiring further corrections.
Nowadays, all things considered, the author prefers instead a manual computer-aided procedure: in a suitably organised spreadsheet, for each string the user manually tries different alloys and sizes, reading in the charts the pulls and checking how they fitor elseinside the historical envolvents. By trial and error it is not difficult to get eventually a pull curve that follows historical shapes and boundaries, while stresses are kept within safe limits, as shown in Figs. 5 and 6 above.
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