|Guitar strings from the nineteenth century to the advent of Nylon - A Brief Explaination of the Physics of Strings|
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A BRIEF EXPLANATION OF THE PHYSICS OF STRINGS
When a string made from any material is progressively stretched between two fixed points (which determine its vibrating length), at a certain moment it reaches a frequency at which it breaks. This point corresponds to the breaking load of the string, which in the case of gut is about 32 kg/mm². The value of this limit frequency, known as the 'breaking frequency', is completely independent - strange as it may seem - of diameter, as may be easily demonstrated either mathematically (applying the general formula for the strings) or experimentally. This limit frequency is in direct proportion to the vibrating length of the string. In other words, the product of the vibrating length - in metres - and the breaking frequency - in Hz - is a constant defined as the 'breaking index'. The average breaking index of a modern gut string in experimental conditions is 240 Hz/m, obviously corresponding to a breaking load of 32 kg/mm². This means that at a vibrating length of a metre the string will break, theoretically, at 240 Hz. If one divides the breaking index by the tuning frequency chosen for the first string, this will produce the vibrating length at which the string will break. For the first string of a baroque guitar in E at the supposed seventeenth-century tuning standard of A = 415 Hz (according to which E = 315 Hz), the theoretical length at which the first string will break is 75 cm; the choice of a 'working' vibrating length will therefore have to consider a prudential shortening of this limit length. But by how much? To answer this question we must return briefly to the period preceding the advent of overspun bass strings. Musicians had always known that a string works best when, subjected to what seems to be the right degree of tension - that is to say, neither excessively taut nor excessively slack to the touch - it has the smallest possible diameter. Once this degree of tension was established, it then had to be distributed evenly across all the strings of the instrument. It was known, moreover, that as the section of a string increased - its tension and vibrating length being equal - it reached progressively lower frequencies, but that at the same time its total acoustic output (in terms of dynamics, the richness of overtones and the duration of the sound) diminished, to the point of becoming - above certain diameters - practically unacceptable. The only solution possible at that time - they were basically limited to gut - was to increase the vibrating length up to the physical limit determined by the first string, as seen above. Only in this way could one hope to reduce the diameter of all the strings as much as possible (particularly the bass strings, which were the thickest and therefore the most critical), thereby drawing from them their best sonority. Vibrating length and diameter are in fact inversely proportional. On the basis of the vibrating length in surviving plucked string instruments and the mechanical properties of gut strings, researchers have speculated that the working length probably entailed a prudential shortening of the hypothetical 'breaking' vibrating length by about 2-4 semitones. Thus, the above-mentioned vibrating-length limit for the guitar - 75 cm - corresponds to a 'working' length of about 69 cm, which is in fact very typical of surviving five-course instruments. With the appearance of overspun strings, the rule of increasing the vibrating length as much as possible no longer applied, in that the acoustic exuberance of the new bass strings was such as to recommend to eighteenth-century instrument makers a salutary shortening of the vibrating length (about two frets less) so as to increase agility of performance. It goes without saying that because the vibrating length was shortened and the tensions remained the same, the diameters of the first three strings, made purely of gut, had, by the laws of physics, to be increased, inevitably resulting in a certain loss of brilliance and a 'sweeter' sound: more viola than violin, as it were.