Below sixteen and one-half vibrations per second, and above four thousand seven hundred and fifty-two,--as high, in fact, as forty thousand,--sounds are audible, but not musical, being either too low and throbbing or two high and piercing to be agreeable. In all countries this range of musical sounds is divided into octaves,--the octave of any given note having simply double the number of air-vibrations. At the present time, in the Western world, each octave is divided, as every one knows, into twelve notes, indicated on the piano by the seven white keys and the five black. For instance, the middle C of the piano has two hundred and sixty-four vibrations per second, the C above has, of course, just double, or five hundred and twenty-eight vibrations per second. In the chromatic scale these two hundred and sixty-four vibrations, which make the octave, are divided into only twelve intervals, an average of twenty-two vibrations to the interval. In the octavo above the average would be twice that, or forty-four, and so on doubling to the end. There is a change in pitch with the addition of so much as a fraction of a vibration per second. As a matter of fact, musicians can detect the variation of pitch caused by the difference of half a vibration per second in the middle octaves; the power to detect changes in pitch due to fractional changes in vibrations decreasing towards the bass and treble. With this power of discriminating a thousand degrees of pitch in a single octave the Western world is content to arbitrarily and mechanically divide the octave into but twelve tones and semi-tones. The Arabic octave contains twenty-four quarter-tones, and Oriental nations generally take cognizance of intervals so small they seem to us discords. Helmholtz requested a distinguished musician to investigate this matter in Cairo, and this is the report: “This evening I have been listening attentively to the song on the minarets, to try to appreciate the quarter-tones, which I had not supposed to exist, as I had thought that the Arabs sang _out of tune_. But to-day as I was with the dervishes I became certain that such quarter-tones existed, and for the following reasons: Many passages in litanies of this kind end with a tone which was at first the quarter-tone and ended in the pure tone. As the passage was frequently repeated, I was able to observe this every time, and I found the intonation invariable.”[35] All of which goes to show how susceptible the highly-trained ear is to fine gradations and combinations of sound and how easy it is to become accustomed to coarse intervals when the finer are no longer used. * * * * *