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
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 | C¹ |
| 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 C1, 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.