MUSIC 79 In the minor mode we are often obliged to elevate by a semitone the seventh and also the sixth note of the gamut. To obtain absolute purity, all gamuts on an instrument of fixed sounds, like the organ or piano, would require an extraordinary, indeed an almost impracti~ cable complication. Mr. A. J. Ellis has shown in a paper published in the " Proceedings of the Royal Society," vol. xiii., " On a perfect Musical Scale," that within the compass of an octave 72 notes would be required to give an absolutely perfect command of all the keys that are now used in music. It has there- fore been found necessary to make a compro- mise, in perfect harmonious effects, in the con- struction of instruments with fixed sounds; and thus has come about the universal adop- tion of the musical scale known as that of "equal temperament," so called because be- tween any two contiguous notes the same in- terval (called a semitone) exists throughout the whole scale. As the octave is divided into 12 equal intervals, it follows that each of these intervals is equal to |/2, or to 1-05946. This scale being a compromise, the major triads are slightly dissonant. Thus, in the natural scale the ratio of the vibrations of G : E : G are as 1 : 1*25 : 1*5 ; but on the scale of equal temper- ament these same notes bear to each other the vibration ratios of 1 : 1-2599 : 1-4983. Thus it follows that the interval of the major third is sharpened, while the fifth is flattened. If we take the middle octave of the piano for an ex- ample, we shall find that E and A are three vibrations a second too sharp, while the fourth and fifth are out of tune by one vibration a second. For convenience of comparison we here give the two scales. The natural scale is placed below the scale of equal temperament. The numbers of vibrations in a sound, correct to the nearest unit, are written under the notes. When the vibration number is a fraction more or less than the number given, the sign + or is respectively attached to the number. The notes belong to the middle octave of the piano. C Ctf D Dfl E F F| a Gtf A Aft B 264 280- 296+ 314- 333- 352+ 373+ 395+ 419+ 444- 470+ 498 +
264 D E|, E F 297 317- 330 352 G Aj, A 396 422+ 440 Tb B 469+ 495 The ratio of the semitones of the tempered scale is approximately |f , and a tone on this scale barely differs from the major tone of -f. This invention has been variously attributed to Keidhart and Werckmeister, to Sebastian Bach, and to Lambert the geometrician. This musi- cal scale was first applied to the clavichord, and Emanuel Bach, son of Sebastian, said a well tuned clavichord was the most accurate of all instruments ; this remark is readily un- derstood when it is explained that, from the manner of production of the sounds on this instrument, the higher harmonics, even when evolved, are feeble and soon die out from the sounds, while the resultant tones appear only at the moment the chords are forcibly struck. But all organists know how harshly intervals are given on a stop of reed pipes, or on the furniture register, tuned to the equal-tempered scale. This harshness is due to the imperfect tuning causing the beating of harmonics and resultant tones. An excellent method of com- paring the relative effects of natural and of tempered tuning is to listen to a few voices singing a series of sustained chords of three or four parts without accompaniment, and then 1! 3ten to exactly the same chords with the ac- )mpanimeht of a piano or melodeon. In the itter case the harshness of the accompaniment is forcibly brought out. One naturally sings irfect intervals, and a violinist with a refined IT will involuntarily play on the natural scale ; but if the voice is educated by the accompani- lent of the piano instead of the violin, and if violinist is always accompanying the fixed 533 VOL. xn. 6 tones of an orchestra, then they will both have acquired the habit of rendering the false in- tervals of the tempered scale. The vibration fraction of an interval expresses the ratio of the numbers of vibrations performed in the same time by the two notes which form the interval. Thus, the vibration fraction means that while the lower of the two notes, forming a major third, makes four vibrations, the higher of these notes makes five. Therefore, while the lower makes one vibration, the higher makes five fourths of a vibration, or one vibra- tion and a quarter. Conversely, while the higher note makes one vibration, the lower makes four fifths of a vibration. This reason- ing is general, and hence follows this rule : Any fraction greater than unity denotes the number of vibrations, and fractions of a vibration, made by the higher of two notes forming a certain interval while the lower note is making a sin- gle vibration. Similarly, any fractioif less than unity indicates the proportion of a whole vibration performed by the lower note while the upper is making one complete vibration. The rules for adding and subtracting musical intervals are as follows : To find the vibration fraction for the sum of two intervals, multiply their separate vibration fractions together. To find the vibration fraction for the difference of two intervals, divide the vibration fraction of the wider by that of the narrower interval. Thus, a major third added to a fifth gives a major seventh ; while a major third subtracted from a fifth leaves a minor third. One of the most common applications of the second' rule
