Several of these intervals or ratios of frequencies have received names. Thus the interval C to E, = 4:5, is called a _major third_, and the interval E to G, = 5:6, is called a _minor third_; the interval C to G, = 2:3, is called a _fifth_, and that of C to C^1, = 1:2, is called an _octave_. For the purposes of music it has been found necessary to introduce other notes between the seven notes of the octave. If a note is introduced which has a frequency greater than any one of the seven in the ratio of 25 to 24, that is called a _sharpened_ note; thus the note of which the frequency is ³⁄₂_n_ × ²⁵⁄₂₄ would be called _G sharp_, and written G♯. In the same way, if the frequency of any note is lowered in the ratio of 24 to 25, it is said to be _flattened_. Then the note whose frequency is ³⁄₂_n_ × ²⁴⁄₂₅ would be called _G flat_, and written G♭. It is obvious that if we were to introduce flats and sharps to all the eight notes we should have twenty-four notes in the octave, and the various intervals would become too numerous and confusing for memory or performance. Hence in keyed instruments the difficulty has been overcome by employing a _scale of equal temperament_, made as follows: The interval of an octave is divided into twelve parts by introducing eleven notes, the ratio of the frequency of each note to its neighbours on either side being the same, and equal to the ratio 1 to 1·05946. The scale thus formed is called the _chromatic scale_, and by this means a number of the flats and sharps become identical; thus, for instance, C♯ and D♭ become the same note. The octave has therefore twelve notes, which are the seven white keys, and the five black ones of the octave of the keyboard of a piano or organ. Every one not entirely destitute of a musical ear is aware that certain of these musical intervals, such as the fifth, the octave, or the major third, produce an agreeable impression on the ear when the notes forming them are sounded together. On the other hand, some intervals, such as the seventh, are not pleasant. The former we call _concords_, and the latter _discords_. The question then arises—What is the reason for this difference in the effect of the air-vibrations on the ear? This leads us to consider the nature of simple and complex air vibrations or waves.