Contributions to the History of Musical Scales

gs of regularly decreasing length; here, of course, difference of tension may nullify the scale due to the lengths. One form is shown on Plate 8. 2. Pan's pipes are sometimes seen with regularly decreasing lengths; it is true that this regularity is not very common, but it is the only principle of scale building (except the Chinese cycle of fifths) yet recognizable in these primitive instruments. (Plate 9.) 3. Instruments of the bar type are found frequently in our orchestras and bands under various names, as xyloplume; they are familiar in children's toys and are widely distributed in savage and half-civilized lands under the names'of marimba, bal a fang, harmonicon, etc. (Plate 8 and fig. 8.) The law of the uniform bar is that the frequencies of vibration of a series of bars of the same material are proportional to the quotients of the thickness divided by the square of the length; the breadth is immaterial if it is uniform. So if one takes a series of uniform bars of the same thickness and regularly decreasing length he may obtain a series of ascending notes. Thus, let the first bar be 24 units long (for example 24 cm.), the successive bars decreasing by one unit; the eighth bar will be 17 units long, and the fifteenth bar 10 units; the series of frequencies would then be as the reciprocals of the squares of 24, 23, etc., so giving to the ear a series of increasing intervals; with these proportions bar No. 8 would give the Octave of the first, but...