Page:The American Cyclopædia (1879) Volume I.djvu/90

 70 ACOUSTICS the friction causing a vibration of the air column in the pipe, on the same principle as the friction of a violin bow causes the vibra- tion of a string. The length of the wave pro- duced in an organ pipe is equal to the length of the pipe ; and as sound travels through air with a velocity of about 1,090 feet per second, it must pass through a pipe 32 feet long hi nearly the 32d part of a second, and thus produce 32 waves per second. If the pipe is 16 feet long, we must have 64 waves per second; for an 8-feet pipe, 128 waves; 4 feet, 256; 2 feet, 512; 1 foot, 1,024; 6 niches, 2,048; 3 inches, 4,096; and 1 inch, 8,192 waves. These are the correct velocities of vibrations of the tones represented by the note called C, Ut, or Do, from octave to octave, according to the so- called theoretical pitch. In Handel's tune the lower C corresponded to 31 vibrations per sec- ond, and the Italian opera in London had it in 1859 at 34 vibrations ; while the pitch re- cently established by the French conservatory of music and by a congress of musicians in London agreed to nearly 33 vibrations, corre- sponding to the Stuttgart pitch. Only the eight octaves mentioned above are used in mu- sic. The capacity of the ear, however, extends an octave below the lowest and more than two above the highest of these figures, being be- tween 16 and 38,000 vibrations per second; but there is a difference in this regard between individuals, some persons being perfectly deaf for very low or very high tones distinctly heard by others. The seven different tones of the so- called diatonic scale are interpolated between the octaves given above, and expressed by the customary notes and staff* of five lines with clef, or by the letters C, D, E, F, G, A, B, C. They correspond for the lower octave with the ve- locity of vibrations 82, 86, 40, 42f, 48, 54, 60, and 64 vibrations per second respectively ; by multiplying either of these numbers by 2, 4, 8, 16, &c., we obtain the velocities of any other octave. It is seen that some of these numbers bear simple ratios to one another, as C : C = 1:2, C:G = 2:8, C:F = 3:4, C:E = 4:5, E : G = 5 : 6-; these tones harmonize, the others are discordant. The further comparison of the numbers shows that the differences between the 3d and 4th and between the 7th and 8th of the scale are less than those preceding or following. This has given reason for the in- terpolation of five other tones between those of which the differences are greater, so as ap- proximately to equalize these differences; hi this way 12 tones in each octave have been obtained, forming a scale called chromatic. These interpolated tones are inappropriately called semitones, and designated with the same sign as the next note, but preceded by a $ (sharp) or -, (flat). This scale is represented in the velocity of vibrations and in name as follows : 84 88 45 51 67 82 86 40 42% 48 54 60 64 Ct Dt Ft Gt A| C D E F GAB C The keyed instruments give a material repre- sentation of this scale. The relation of pro- gression between its tones, when tuned accord- ing to the proportions given here, is so irregu- lar, that when transposing the diatonic scale, that is, when commencing it at another tone than C, very impure harmonies are obtained. This is corrected, or rather compromised, by making the mutual proportions of the 12 num- bers representing the chromatic scale such as to obtain a regular geometrical series ; this is the so-called equal temperament. In order to accomplish this with strict mathematical accu- racy, we have only to interpolate 11 terms of such a series between the numbers 1 and 2, which express the relations between a tone and its octave ; this is mathematically ex- pressed by the series 2, 2^, 2"&, 2^, 2~&, fec., to 2^; or by logarithms: log. $, log. f|, log. fj, &o., to log. f, which by calculation gives the series 1-000, 1-0594, 1-1225, 1-1892, 1-2599, 1-3348, 1-4142, 1-4983, 1-5874, 1-6818, 1'7818, 1-8877, 2-000. Multiplying each of these numbers by 32, we obtain the velocity of vibration for the low- er octave, for the absolute equal temperament : 88-8908 88-0544 82-000 85-9200 40-8168 42-7186 C D E 45-2544 50-7968 57-0176 47-9456 58-8176 60-4064 64-000 Ft Gt At G A B C It is seen, by comparison with the numbers mentioned before, that this series gives Ct, D, G, and G| too low, while the other eighth tones are too high. However, this is only the case when considering the interpolated semitones as sharps; but as we must use Ct for D[>, Dt for E|,, &c., and the calculation for the tones corresponding with these flats gives us differ- ent figures, between which and the former the equal temperament is a compromise, the ad- vantages are acknowledged to be with the lat- ter, and it is now therefore universally adopted. (See Music.) A column of air in a pipe will not necessarily vibrate in such a way that each wave will be equal to the length of the pipe. By modi- fying the manner of admitting the air, either by increased pressure or changing the aperture, the waves may be made one half, one third, one fourth, one fifth, &c., of the length of the pipe. In this way the so-called harmonics and the tones of the French horn are pro- duced. They are called over-tones, if the fundamental vibration producing the lowest tone is still heard at the same time. In order to produce all kinds of shorter waves ^by means of the same pipe, holes may be made h> its sides, closed by the fingers or by proper vaVes. The opening of these holes is nearly equivalent to a shortening of the pipe. Thus the different tones of the flute, clarinet, haut- bois, bassoon, and several other wind instru-