It will be found of course, as must be apparent to the reader, that the differential factor here suggested does not provide a complete solution to the problem of allowing for the exhibited differences between theory and practice. It does, however, provide a true guide to the lengths. There is of course a difference of produced frequency to be allowed for yet. Fortunately, however, this is provided for by the graduated thicknesses of pianoforte wire. By taking advantage of this almost geometrically proportioned graduation of diameter we are able to calculate a stringing scale that, if adhered to, will give the nearest possible approximation to complete harmony between theory and practice. That is to say, we can proceed with a string-length calculation based upon the differential factor already obtained, and then by arranging the distribution of the string thicknesses according to the diameters that are provided by the manufacturers of music wire, we may obtain a true estimation, not only as to the thickness of wire to be used at each place, but also as to the lengths proper to each string. Of course the reader will remember that the matter of pitch is of considerable importance in all calculations of this kind. A difference in pitch implies difference of tension when the other factors remain equal, and we therefore have calculated the following tables on the assumption that the pitch to be used is that known as the International or C 517. Attention is, therefore, directed to the following
TABLE SHOWING TRUE LENGTHS OF OCTAVE STRINGS FROM THE HIGHEST C STRING TO THE LAST C STRING THAT IS USUALLY LEFT UNWRAPPED.
C⁵ = 2.048 in. = 2-1/25 + ... Approx. C⁴ = 2.048 × 1.9375 = 3.968 in. = 3-24/25 + ... " C³ = 3.968 × 1.9375 = 7.688 in. = 7-17/25 + ... " C² = 7.688 × 1.9375 = 14.875 in. = 14-7/8 + ... " C¹ = 14.875 × 1.9375 = 28.820 in. = 28-4/5 + ... " C = 28.820 × 1.9375 = 55.828 in. = 55-4/5 + ... "
[NOTE.--The length of the first string is chosen arbitrarily, but as given is a very close approximation to the practice of the best American makers. The vulgar fractions are calculated from the decimals and the error in no case exceed about one-fiftieth of an inch. The differential factor is, as we know, 1/16. Therefore we multiply by (2 [-] 1/16) or 1-15/16; in decimals 1.9375.]