Bleyer's article appeared in the
published in Leipzig. The article discussed many aspects of the upright fortepianos of the firm Wachtl und Bleyer in Vienna. The portion of the text referring to wire:
We then set about improving these instruments, with measured steps, but always progressing forward. The most essential [improvement] was to give the thicknesses of the strings a proper proportion; for he who places his trust and belief in the wire manufacturers is often scandalously deceived. Not through any shortcoming in their skill, no, but rather because unbeknownst to their customers, one often finds two numbers with the same diameter and one number with two different diameters. Furthermore one is easily convinced that not all manufacturers adhere to the same unit of measure. We gave our fork-shaped wire gauge the following arrangement: between two strings a & b which have diameters related to one another by 1:2, 15 steps are interpolated, and in such a manner that when one arranges all the string diameters in the proper order, a geometric series appears. The thickness of the strings must increase and decrease in a geometric relationship if the tone of the instrument is to sound even. Thus we have 17 numbers from a to b. The locally available strings, like the Nürnberg strings, only have 6 numbers between a and b, and if one interpolates half numbers, one has only 15 numbers, the half numbers of which can easily lead to errors. (trans. P.P.)
From Bleyer's "snapshot" of the Viennese piano making at the time, we can draw the following conclusions:
It is most instructive that Bleyer appears to make a distinction between methods used in sorting the wire and the actual diameters available. For example, he says one can "interpolate" half sizes to get twice as many steps. Obviously, if half sizes were readily available from the suppliers, Bleyer's whole complaint about the coarseness of the system and the remark about "interpolating" would have no relevance. Bleyer also says nothing about having wire specially drawn for his 16th root of 2 system; evidently he felt this degree of fineness was needed to accurately sort the wire which was ostensibly drawn according to the 7th root of 2 "Nürnberg" system, which was not only much coarser, but was also based on an incommensurable drawing ratio. This implies that the range of diameters encountered was almost infinitely variable, and that the original reduction scheme used by the wire manufacturer was of no importance to the end user. Logically we can conclude that Bleyer's intent was not necessarily to find 17 different diameters that would exactly or nearly exactly fit each one of his 17 gauges, but simply to refine the increments of a measuring system used to sort whatever diameters were encountered. If this were indicative of a more widespread approach among piano makers, one would expect to find occasional instances of a single diameter, which fell near the border line between two sizes, used in the ranges on a piano marked for both sizes; not surprisingly, this is exactly what we do find in some later instruments (see the published version of my article for examples).
Bleyer's intellectual approach and methodology is precisely the same manner by which organ builders scale the diameters (and other dimensions) of organ pipes. At that time, such methods were already hundreds of years old. The delineation of a space with two lengths which have the proportion 1:2 and the subsequent division of the space into equal logarithmic steps/lengths is an eminently practical approach, since the system can easily be extended beyond the range of initial calculation merely be doubling or halving the units already determined. This approach is so deeply ingrained in organ builder's traditions that they describe the basic scales of different organ stops as "halving (or doubling) on the 17th (or 15th, 16th, 18th and even 19th) pipe". Note that since the beginning pipe is not defined, exactly like Bleyer's undefined diameters "a & b", the terminology inherently describes a logarithmic progression, for only with such a series is it possible to start on any pipe, count up or down n times, and always end up on a pipe with dimensions half or twice those of the starting pipe.
We should not overlook the fact that the scaling of string diameters and the scaling of organ pipe cross-sectional areas has precisely the same acoustic function: scaling the mass of the vibrating medium. Considering the close alliance between the trades of organ and other keyboard instrument making, it is not surprising that piano makers would approach the problem of diameter scaling using the same intellectual and practical tools and methods. Many piano makers, such as Silbermann, Stein, and Walter, were also organ builders, and Bleyer's claim to having been the first to use interlocking layered construction for piano frames indicates he was familiar with organ case construction. Thus Michael Latcham's evaluation that Bleyer's system is a "rationalization" of the Nürnberg system (based on a comparison of his system with Thomées 1866 undocumented report of Nürnberg diameters) and "perhaps reflects a new emphasis [on] or belief in 'scientific' principles and some dissatisfaction with a reliance on the craftsman's tradition" is probably diametrically opposed to the reality of the situation. Furthermore, his implication that Bleyer's belief in the superior evenness of tone achieved with a string sorting scheme based on a logarithmic progression was little more than unfounded faith in mathematical principles, is at odds with modern psychoacoustic knowledge. The peculiarities of the function of the human aural system and the relationships between mass and acoustic power are all best described by logarithmic formulas. Thus Bleyer merely used mathematical terms to describe long-established "craftsman's traditions" which were empirically derived, not theoretical.
Shown here are two such ancient craftsman's traditions, methods used for calculating logarithmic organ pipe scales using nothing more than simply drawing tools: a ruler, a compass (or trammel), and a square. These drawings illustrate a "geometric" calculation of a scale which halves on the 16th pipe, using both of the possible solutions: (1) constructing a regular progression of dimensional change (i.e., a sloping line) and dividing this line into logarithmic steps; or (2) demarking regularly-spaced steps and constructing a logarithmic progression (i.e., a curve) to fit the steps.
The first method involves the drawing of a right triangle with a specific proportion between the base (AB) and altitude (BD). The base length is then swung up to the hypotenuse with a compass and a new point E is marked. From point E a new altitude is dropped to the base, defining point C. This process is repeated, and if the initial proportion is correct, the desired nth step (marked F) will fall exactly at the half length of all three sides. Once the point F is reached, the next smaller step can be ruled on the base line simply by taking half of the first step (BC); in a similar manner the calculation can be extended to larger dimensions by extending the base and hypotenuse lines beyond the points B and D and doubling the step sizes beginning at F. Once the desired logarithmic division of the base line is accomplished, other dimensions can be derived by striking new hypotenuse lines at other angles.
The proportions required for each standard scale were probably first determined by trial and error, and then became part of the oral tradition passed from master to apprentice, represented in easy-to-remember integer values. For example, a Pythagorean scale (12th root of 2) is constructed with an altitude:base proportion of 7:20; Bleyer's 16th root of 2 (which organ builders would have called "halving on the 17th pipe") with 3:10; and the original "Nürnberg" system, halving on the 8th step, with 7:15.
The other ancient method is even easier, requiring no knowledge of underlying proportions. One simply marks out a series of evenly spaced lines, and marks off the desired nth line three or more times. A starting length is ruled out on the 1st line, and this length is successively halved at each nth line (marked with "x" in the drawing). Then a thin batten of wood (called a spline) is bent so as to intersect the x's, along which a "spline curve" is drawn. The intermediate lengths are then taken directly from the drawing. Three points are needed to accurately bend the spline, but once these two groupings of n steps are drawn, shorter or longer lengths are derived by simply halving or doubling.
With either method, the template need only be drawn once. The calculation of values other than those actually used in the construction of the triangle or curve can be derived by interpolation. For example, if one wanted to know the depth of the tenth pipe in a scale halving on the 16th but starting with a dimension half way between BD and CE, one would simply count down 10 steps and take the length half way between those two lines. The same method of visual interpolation works incredibly well with the spline curve. Actually, one need really only draw one template, perhaps a Pythagorean 12-step, for both templates can also be used to calculate any other divisions, even highly irrational progressions, such as scales which halve on "n + (a fraction)-th" step. This is accomplished with canted sticks, marked out from the horizontal scales and then laid at angles upon the templates to redefine the horizontal increments. The resulting scales can often only be described by quite sophisticated mathematics (such as Bleyer's recommended octave proportion of 1:1.9458608), even though they were created with the simplest of methods. Such calculating templates can be constructed in minutes, used for decades, and yet have no real value other than the wood upon which they are drawn. Thus we would not expect to find them listed in workshop inventories of deceased instrument makers.
Both drawings are from F. E. Robertson's 1897 "A Practical Treatise on Organ-Building". This little-known though excellent work discusses the history of organ building theory in detail, and the bibliography cites 299 titles, most in German, going back to the 17th century. Though Robertson is fluent in the use of exponential calculation and provides many tables of logarithmic scales in the appendices, he refers to the scale templates as being "better still", since one can take the working dimensions directly from the template rather than calculating.