|Loading of Gut - Bridge Holes|
1. Bridge holes
The hypothesis that the loading of gut could have been the common and relevant characteristic of bass strings employed between 16th and 18th centuries found its way in the light of some data recording, which afterwards became systematic, of bridge holes sizes from historic lutes built before, or very close to 1664, year to which the earliest surviving historic report of overspun strings date back to (9). It should be underlined, however, how scant and scattered the spreading of the new type of strings must have been, probably on the ground of the habit musical taste, and the fact that traditional bass strings in the face of the novelty, kept being competitive. T. Mace (10) over ten years after J. Playford's announcement, make no mention of them. J. Talbot (11) by then prefers to use conventional bass strings, and so does L. Mozart who well into the 18th century, implicitly excludes, in his 1756 violin tutor, the use of overspun strings.
For the lute the bond to old habit remains very strong. Gut bass strings of "... a deep dark red colour ..." (12) can be seen in a portrait by the berliner painter A. Pesne (1678-1758) painted around 1740-2 and now in the Schloss Charlottenburg, Berlin, portraying the graceful Eleonore von Kayserling playing a 13 courses German theorbed lute, probably tuned in D minor. Thus, on the ground of such observations, some of the original instruments examined are from a few decades after the watershed implicitly established by Playford's announcement.
In practical terms the lowest courses' hole diameters in original instruments bridges were found to be so small as to allow no all-gut string passing through them a sufficient working tension and therefore a sufficient sonority, a sonority moreover only granted to strings possessing a high degree of elasticity such as rope structured ones for instance, since plain gut strings, even at working tension considered correct, proves to be totally unusable. In table 1 some significant examples of very accurately measured instruments are recorded. These lutes were presumably tuned in renaissance tuning, possible doubts or further remarks are recorded aside. The low strings holes were measured by means of rods of increasing known diameters, thus the maximum passing diameter was found. When examining these data it must be borne in mind that we are dealing with bridge holes diameters and not strings diameters, which were bored by the lute maker with some amount of empirical oversize. In spite of this handicap, the results are absolutely surprising.
Before examining the data in the tabs, it will be useful to briefly remind some considerations, quite often treated anyway, on the physics of strings (2)(3). Breaking point define the tension at which a string of 1mm in section breaks for modern gut. This is estimated at about (13). At equal vibrating length, gut strings of different diameters will break, theoretically, at the same frequency (breaking frequency) (14). Through the general equation for strings, it can be inferred that the product of this critical frequency by the string vibrating length (in meters) is a constant K (breaking index). Such relation when expressed in other terms ( ) allow to quickly calculate the theoretical upper limit at which the cantino of each historical instruments examined will break.
The common procedure for tuning a lute (or a viola) for solo playing in 16th century was, as well known, to tune up the treble, empirically, to the highest possible pitch, stopping short of breaking (i.e. before reaching its breaking point). In other words one aimed for the highest working frequency a given string vibrating length will allow. Keeping slightly below breaking frequency, ancient musicians obviously aimed for an acceptable compromise between the top string's playing life and the safeguard of the best acoustic performance of the all-gut lower registers. For instruments with a defined nominal pitch-tuning it is to be expected that the procedure could have taken a perfectly specular route, exploiting, in practice, the maximum string vibrating length which the breaking point, or rather the breaking frequency allowed. Thus at playing pitch the treble worked in any case close to the breaking. Such a hypothesis finds confirmation in the estimates made on the lute depicted in M. Pretorious "Syntagma Musicum", Wolfenb√ľttel 1615-20, where the product of frequency by vibrating length is (15).
Basing themselves on these concepts several scholars today suggest that working at 2-3 semitones below breaking frequency may be considered a reasonable choice (2)(16). For our table it was decided to work at two semitones below the breaking frequency of each historic instrument. Bearing in mind that the internal between courses are constant, it was the possible to assign each course a definite frequency. It is worthwhile to remind that under these working condition a chantarelle, no matter what its breaking is, exploits some 80% of its total strength "reserve".
In absolutely empirical terms, it may be assumed that at more then 4kg tension, at the strings lengths examined, a string becomes too stiff to sound, played with finger pads, with "grace" and "full pearls" (17). At tensions lower than 2.5 Kg the "focus" of the note produced (basic frequency with its baggage of higher harmonics) becomes decidedly imprecise, especially with fairly thick gut strings, where the inner damping coefficient is rather pronounced. The treatises of the past confirm for their part that the strings should feel neither too stiff nor to slack to the touch (15)(18); for sheer convenience, the value of 3 Kg was chosen, a working tension at which there is a tendency to play today.
As far as specific weight is concerned, two different ones where considered: that of usual plain gut (about ) which although lacking any practical application for the manufacturing of bass strings possesses the highest density value for a natural product, beyond which it is only possible to go by means of appropriate treatments. The other is the mean apparent specific weight of a rope-construction string, which represent, in fact, the only model devised today as a possible technological solution employed in the past, and whose density is around .. By processing these parameters (density, string length, frequency) (19), it was possible to tabulate the following:
Column A: the diameter (in mm) which a traditional plain gut string (a) or a rope-construction one (b), at 3.0 Kg tension should possess in order to produce the corresponding working frequency, assuming a treble, working at two semitones below breaking point.
Column B: the specific weight which strings passing through each bridge hole, at 3.0 Kg tension, should possess for a working frequency assigned to bass, according to the above mentioned criteria. As can be easily noticed, the densities so obtained are rather higher than those of the two types of strings considered.
Column C: the tension in Kg each type of strings (a or b) would have at diameters equal to those recorded from original bridges. This last column looks rather interesting, since the only significant variable is represented by the breaking point of gut, which in turn determines the working frequency to assign to each bass course.
For sheer curiosity we have considered also the mean value of the breaking point of specimens of current commercial strings (sheep and beef) which is 34 (they range between 31 and 38 ) but the resulting working tensions were for the most part below any acceptable value. Finally, the hypothesis was considered that treble strings of the 16th and 17th centuries may have been remarkably stronger than those available today, in order to allow, at equal vibrating length, a higher pitch and therefore move the working tensions up to a range of values which could be judged acceptable. Through the string equation, it was estimated that it would be necessary to reach a breaking point of at least 40 to 48 , which seems frankly excessive.
Being the product between the vibrating length and the breaking frequency (or the working frequency) for the cantino a constant, we can assume that the same product for each course is a constant as well, so the knowledge of the vibrating length becomes inessential to determine some important parameters as the specific weight and the working tension. Thus is sufficient to know the bridge hole diameter to obtain what is necessary, and is no more a possible source of doubts the possibility that in the past some bridges of the historic instruments observed could have been moved in a new position, different from the original one.
For baroque lutes with a d-minor tuning, the influence of the breaking point of gut on the data processing looses completely significance. The top string of these instruments is always nominally defined as f', whose actual frequency depends solely from the pitch standard in use, for example kammerton for German lutes in the 1st half of the 18th century (20). In table 2 examples are recorded of this type of lutes; the method used is substantially different from the preceding table, owing to the fact of being able to define more accurately the nominal pitch of each course.
In order to be able to assign them an accurate working frequency, the whole range of frequencies was taken into account, which scholars have considered as valid pitch standards in use in Europe in the 17th an 18th centuries, with particular attention to France and Germany; a range of frequencies, which is in any case included, between a minimum of zero semitones (0s) and a maximum of two semitones (2s) below our present standard of 440 Hz (21)(22).
A simple check using the method of table 1, suggests anyway to exclude some values in the higher side (0s, and 1s for vibrating length greater than 0.71 m). With the vibrating length of baroque lutes we would have a breaking frequency for the top string too close to these values. Thus parameters computed for a frequency standard referred as 0s (and 1s for vibrating length greater than 0.71 m) assume a purely speculative meaning.
The great importance of this table must not be overlooked, owing to its being in no way dependent on the breaking point of gut, a fact which drastically reduces the degree of uncertainty of the processed data.
Column C for instance, is only conditioned by the standard pitch assumed as basis. The specific weight used for these calculations is that of a rope-construction string. The information given by Baron (top string equal kammerton f') suggests some considerations on the breaking point of treble strings in the 18th century. A large number of surviving instruments from that period have a vibrating length of between 0.70 to 0.72 m Assuming a top string working at two semitones below breaking point (by which it should last 3-4 weeks before breaking, in accordance with both modern and Baron's indications) at the frequency range hypothesised by scholars for German kammerton (a' = 440 √∑ 390 Hz) it is simple to calculate a breaking point "window" of 32√∑36 (23), that is remarkably close to those measured in modern commercial gut strings.
By following the same procedure it is possible to try and work an estimate of the breaking point of the treble strings of the 16th and 17th centuries. Several lutes built in Venice in the last 16th and the first half of the 17th centuries, surviving today, have a string vibrating length of between 0.57 to 0.59 m (24). With two semitones margin below the breaking frequency limit suggested as venetian pitch standards of that period (450 √∑ 466 Hz), which seems to have kept rather stable in that time span (21), a range of breaking points is obtained of between 33 to 36 , the assumed pitch being ' for the chantarelle. Assuming an a' pitch instead, the breaking point values rise to 38√∑40 .. These deductions represent yet another positive support to the table 1 data.