=335. Major and Minor Triads.=--The notes C, E, G (_do_, _mi_, _sol_) form what is called a _major triad_. The _relative vibration numbers_ corresponding are 24, 30, 36. These in simplest terms have ratios of 4:5:6. Any three other tones with vibration ratios of 4:5:6 will also form a major triad. If the octave of the lower tone is added, the four make a major chord. Thus: F, A, C´ (_fa_, _la_, _do_), 32:40:48, or 4:5:6, also form a major triad as do G, B, D´ (_sol_, _ti_, _re_), 36:45:54, or 4:5:6. Inspection will show that these three major triads comprise all of the tones of the major scale D´ being the octave of D. It is, therefore, said that the major scale is based, or built, upon these three major triads. The examples just given indicate the mathematical basis for harmony in music. Three notes having vibration ratios of 10:12:15 are called _minor triads_. These produce a less pleasing effect than those having ratios of 4:5:6. =336. The Need for Sharps and Flats.=--We have considered the key of C. This is represented upon the piano or organ by white keys only (Fig. 325). Now in order (a) to give variety to instrumental selections, and (b) to accommodate instruments to the range of the human voice, it has been necessary to introduce other notes in musical instruments. These are represented by the _black keys_ upon the piano and organ and are known as _sharps_ and _flats_. To illustrate the necessity for these additional notes take the major scale starting with B. This will give vibration frequencies of 240, 270, 300, 320, 360, 400, 450, and 480. The only white keys that may be used with this scale are E 320 and B 480 vibrations. Since the second note on this scale requires 270 vibrations about halfway between C and D the black key C sharp is inserted. Other notes must be inserted between D and E (D sharp), between F and G (F sharp), also G and A sharps. [Illustration: FIG. 325.--Section of a piano keyboard.]