267. =The Diatonic Scale.=--In order that you may see the relative numbers of the vibrations for each of the notes I will give them for the whole scale. They are as follows: 1 9/8 5/4 4/3 3/2 5/3 15/8 2 C D E F G A B C. According to this the note D has nine vibrations to every eight vibrations of C, E has five to every four of C, etc., the octave C having just twice the number of vibrations that the fundamental note C has. You have here expressed the _proportion_ between the numbers of vibrations in the different notes. Suppose, then, that you know the number of vibrations in a second that C, the fundamental note, has, you can readily calculate the number of vibrations of each of the other notes. It is done by multiplying the number which C has by the fractions over the other notes. Thus if the number of vibrations in a second in the fundamental note be 128, by this process we make the vibrations of all the notes to be thus: C D E F G A B C 128 144 160 170 192 213 240 256. There are really but seven notes in what is called the diatonic scale, the eighth note, C, being truly the first of seven other notes above, having relations to each other similar to those of the notes below, and constituting another octave. So we may have several octaves, one above another. It is interesting to observe that the proportionate lengths of strings required to produce the eight notes of the scale have an exact numerical relation, but the _reverse_ of that of the numbers of the vibrations. Thus if you have eight strings of the same size, their vibrating lengths required for the notes are as follows: C D E F G A B C 1 8/9 4/5 3/4 2/3 3/5 8/15 1/2. For the notes of the octave above the lengths are thus: C D E F G A B C 1/2 4/9 2/5 3/8 1/3 3/10 4/15 1/4. 268. =Unison.=--In tuning instruments so as to make them harmonize the result is obtained when the corresponding parts of the instruments have the same number of vibrations. Thus the string in one violin that gives any particular note must vibrate just the same number of times in a second that the strings giving the same note in other violins do, or it will not be in perfect unison with them. The same is true of other strings for other notes, and also of the corresponding parts of all kinds of instruments which are to be played together. When, in tuning instruments together, it is said that a string of a violin, for example, is too _flat_, the difficulty is that it does not vibrate with sufficient rapidity, and it is therefore screwed up to make its note _sharp_ enough, as it is expressed, to be in unison with the note of the corresponding strings or parts of other instruments.