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from 1726 organist at Minden and from 1732 at Nordhausen, who in 1738 claimed that he had made models between 1717 and 1721. Neither of these efforts seems to have been connected with Cristofori's or to have had any result. The next practical step was taken by Gottfried Silbermann (d. 1753), a noted organ-builder at Freiberg, who at intervals from 1726 made several pianos on the Cristofori model (mostly, it seems, for Frederick the Great). But it was not till after 1755 and in England that piano-making became a business of importance (see sec. 160).

136. Tuning and Temperament.—The practical problem of tuning all keyboard instruments is an intricate one, unless playing is to be confined to but a single scale or tonality, or unless the number of keys to the octave is many more than twelve. Back of it all lies the question of the true theory of intervals, which began to be discussed six centuries before the Christian era (by Pythagoras). And, however this question is answered, the moment that the fundamental scale is augmented by chromatic tones or the slightest modulation attempted, difficulties begin to multiply. Hence discussions of tuning steadily increased from the 16th century onward. The earliest-known formal system of tuning dates from 1571 (Ammerbach), though this came far from solving the problem.

The Pythagorean theory of intervals, which ruled until the 16th century, rested on the assumption that the perfect fifth (3/2) is the only unit, beside the octave, to be used in laying out scales. The theoretical objection to this is that if, from any starting-tone, like C, a series of twelve fifths is laid out (C-G, G-D, etc., disregarding octaves), the final C will be almost a quarter of a semitone too sharp. The most serious practical difficulty began to be felt as soon as the progress of harmonic feeling revealed the beauty and utility of the major triad, for, to be smooth, this required a major third (5/4) distinctly flatter than the Pythagorean third (81/64). The recognition of this true third (beginning early in the 16th century) gave a major scale with the ratios, 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2, that is, one made up of three exactly similar triads, 1-3-5, 5-7-2, 4-6-8 (e.g., C-E-G, G-B-D, F-A-C).

But this perfected scale, admirable as it was for diatonic harmony without modulation, proved difficult to use in determining chromatic tones; for if F#, for example, was taken as the third of D in the triad D-F#-A, then A proved to be flat by a 'comma' (81/80), and if Bb, similarly, was deduced from F, then D was a comma sharp, and so on. Modulation by one remove in either direction always made at least one tone in the new scale slightly false, and every further remove made matters steadily worse. Any attempt, therefore, to tune a keyboard instrument, like the organ or the harpsichord, by first making some one scale true and then making some other scale also true and then another, broke down at the first step and ended in total confusion.