We can, in the next place, pass on now to discuss some matters connected with the theory of music. When regular air-vibrations or wave-trains fall upon the ear they produce the sensation of a musical tone, provided that their frequency lies between about 40 per second and about 4000. The lowest note in an organ usually is one having 32 vibrations per second, and the highest note in the orchestra is that of a piccolo flute, giving 4752 vibrations per second. We can appreciate as sound vibrations lying between 16 and 32,000, but the greater portion of these high frequencies have no musical character, and would be described as whistles or squeaks. When one note has twice the frequency of another it is called the _octave_ of the first. Thus our range of musical tones is comprised within about seven octaves, or within the limits of the notes whose frequencies are 40, 80, 160, 320, 640, 1280, 2560, and 5120. These musical notes are distinguished, as every one knows, by certain letters or signs on a _clef_. Thus the note called the middle C of a piano has a frequency of 248, and is denoted by the sign [Illustration] The octave is divided into certain musical _intervals_ by notes, the frequencies of which have a certain ratio to that of the fundamental note. This ratio is determined by what is called the _scale_, or _gamut_. Thus, in the major diatonic natural scale, if we denote the fundamental note by C, called _do_ or _ut_ in singing, and its frequency by _n_, then the other notes in the natural scale are denoted by the letters, and have frequencies as below. _do_ _re_ _mi_ _fa_ _sol_ _la_ _si_ _do′_ C D E F G A B C^1 n ⁹⁄₈n ⁵⁄₄n ⁴⁄₃n ³⁄₂n ⁵⁄₃n ¹⁵⁄₈n 2n Hence if the note C has 248 vibrations per second, then the note D will have 9 × 248 ÷ 8 = 279 vibrations per second. On looking at the above scale of the eight notes forming an octave, it will be seen that there are three kinds of ratios of frequencies of the various notes. (1) The ratio of C to D, or F to G, or A to B, which is that of 8 to 9. (2) The ratio of D to E, and G to A, which is that of 9 to 10. (3) The ratio of E to F, or B to C^1, which is that of 15 to 16. The first two of these intervals or ratios are both called _a tone_, and the third is called _a semitone_. The two tones, however, are not exactly the same, but their ratio to one another is that of ⁸⁄₉ to ⁹⁄₁₀ or of 80 to 81. This interval is called a _comma_, and can be distinguished by a good musical ear.