9. The speaking tone of the average man's voice has 160 vibrations per second. How long are the waves produced by him at 20°C.?
(4) Musical Scales and Resonance
334. A musical interval refers to the ratio between the pitches[O] of two notes as indicated by the results of the siren experiment. The simplest interval, or ratio between two notes is the octave, C':C, or 2:1 (48:24). Other important intervals with the corresponding ratios are the fifth, G:C, or 3:2 (36:24); the sixth, A:C, or 5:3 (40:24); the fourth, F:C, 4:3 (32:24); the major third, E:C, or 5:4 (30:24); and the minor third, G:E, 6:5. The interval between any two notes may be determined by finding the ratio between the vibration numbers of the two notes. Thus, if one note is produced by 600 vibrations a second and another by 400, the interval is 3:2, or a fifth, and this would be recognized by a musician who heard the notes sounded together or one after the other. Below is a table of musical nomenclatures, showing various relations between the notes of the major scale.
Table of Musical Nomenclatures
| Name of note | C | D | E | F | G | A | B | C´ |
| Frequency in terms of "do" | n | 9/8n | 5/4n | 4/3n | 3/2n | 5/3n | 15/8n | 2n |
| Intervals | 9/8 | 10/9 | 16/15 | 9/8 | 10/9 | 9/8 | 16/15 | |
| Name of note in vocal music | do | re | mi | fa | sol | la | ti | do |
| Treble clef. | ||||||||
| [Music] | ||||||||
| Bass clef. | ||||||||
| [Music] | ||||||||
| International pitch of treble clef | 261 | 293.6 | 326.3 | 348. | 391.5 | 435 | 489.4 | 522 |
| Scientific scale | 256 | 288 | 320 | 341.3 | 384 | 426.6 | 480 | 512 |
| Relative vibration numbers | 24 | 27 | 30 | 32 | 36 | 40 | 45 | 48 |
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.
