[O] Pitch as used here, means _vibration rate_.

TABLE OF MUSICAL NOMENCLATURES

---------------------+---+------+------+------+------+------+-------+---- Name of note | C | D | E | F | G | A | B | C'

---------------------+---+----- +----- +------+------+------+-------+---- Frequency in terms of|_n_|9/8_n_|5/4_n_|4/3_n_|3/2_n_|5/3_n_|15/8_n_|2_n_ "do" | | | | | | | | ---------------------+---+--+---+--+---+--+---+--+---+--+---+---+---+---- Intervals | 9/8 | 10/9 | 16/15| 9/8 | 10/9 | 9/8 | 16/15 ---------------------+---+--+---+--+---+--+---+--+---+--+---+---+---+---- Name of note in vocal|do | re | mi | fa | sol | la | ti | do music | | | | | | | | ---------------------+---+------+------+------+------+------+-------+---- Treble clef.

[Music]

Ba.s.s 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 | 24| 27 | 30 | 32 | 36 | 40 | 45 | 48 numbers | | | | | | | | ---------------------+---+------+------+------+------+------+-------+----

=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 ill.u.s.trate 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.

[Ill.u.s.tration: FIG. 325.--Section of a piano keyboard.]

=337. Tempered Scales.=--In musical instruments with fixed notes, such as the harp, organ, or piano, complications were early recognized when an attempt was made to adapt these instruments so that they could be played in all keys. For the vibration numbers that would give a perfect major scale starting at C are not the same as will give a perfect major scale beginning with any other key. In using the various notes as the keynote for a major scale, 72 different notes in the octave would be required. This would make it more difficult for such instruments as the piano to be played. To avoid these complications as much as possible, it has been found necessary to abandon the simple ratios between successive notes and to subst.i.tute another ratio in order that the vibration ratio between any two successive notes will be equal in every case. The differences between semitones are abolished so that, for example, C sharp and D flat become the same tone instead of two different tones.

Such a scale is called a _tempered scale_. The tempered scale has 13 notes to the octave, with 12 equal intervals, the ratio between two successive notes being the v2 or 1.059. That is, any vibration rate on the tempered scale may be computed by multiplying the vibration rate of the preceding note by 1.059. While this is a necessary arrangement, there is some loss in perfect harmony. It is for this reason that a quartette or chorus of voices singing without accompaniment is often more harmonious and satisfactory than when accompanied with an instrument of fixed notes as the piano, since the simple harmonious ratios may be employed when the voices are alone. The imperfection introduced by _equal temperament tuning_ is ill.u.s.trated by the following table:

C D E F G A B C Perfect Scale of C 256.0 288.0 320.0 341.3 384.0 426.6 480.0 512.0 Tempered Scale 256.0 287.3 322.5 341.7 383.6 430.5 483.3 512.0

=338. Resonance.=--If two tuning forks of the same pitch are placed near each other, and one is set vibrating, the other will soon be found to be in vibration. This result is said to be due to _sympathetic vibration_, and is an example of _resonance_ (Fig. 326). If water is poured into a gla.s.s tube while a vibrating tuning fork is held over its top, when the air column has a certain length it will start vibrating, reinforcing strongly the sound of the tuning fork. (See Fig. 327.) This is also an example of resonance. These and other similar facts indicate that _sound waves started by a vibrating body will cause another body near it to start vibrating if the two have the same rate of vibration_. Most persons will recall ill.u.s.trations of this effect from their own experience.

[Ill.u.s.tration: FIG. 326.--One tuning fork will vibrate in sympathy with the other, if they have exactly equal rates of vibration.]

[Ill.u.s.tration: FIG. 327.--An air column of the proper length reinforces the sound of the tuning fork.]

=339. Sympathetic vibration= is explained as follows: Sound waves produce very slight motions in objects affected by them; if the vibration of a given body is exactly in time with the vibrations of a given sound each impulse of the sound wave will strike the body so as to increase the vibratory motion of the latter. This action continuing, the body soon acquires a motion sufficient to produce audible waves. A good ill.u.s.tration of sympathetic vibration is furnished by the bell ringer, who times his pulls upon the bell rope with the vibration rate of the swing of the bell. In the case of the resonant air column over which is held a vibrating tuning fork (see Fig. 328), when the p.r.o.ng of the fork starts downward from 1 to 2, a condensation wave moves down to the water surface and back just in time to join the condensation wave _above_ the fork as the p.r.o.ng begins to move from 2 to 1; also when the p.r.o.ng starts upward from 2 to 1, the rarefaction produced under it moves to the bottom of the air column and back so as to join the rarefaction _above_ the fork as the p.r.o.ng returns. While the p.r.o.ng is making a _single_ movement, up or down, it is plain that the air wave moves twice the length of the open tube. During a _complete_ vibration of the fork, therefore, the sound wave moves four times the length of the air column.

In free air, the sound progresses a wave length during a complete vibration, hence the resonant air column is one-fourth the length of the sound wave to which it responds. Experiments with tubes cf different lengths show that the diameter of the air column has some effect upon the length giving best resonance. About 25 per cent. of the diameter of the tube must be added to the length of the air column to make it just one-fourth the wave length. The sound heard in seash.e.l.ls and in other hollow bodies is due to resonance. Vibrations in the air too feeble to affect the ear are intensified by sympathetic vibration until they can be heard. A tuning fork is often mounted upon a box called a _resonator_, which contains an air column of such dimensions that it reinforces the sound of the fork's sympathetic vibration.

[Ill.u.s.tration: FIG. 328.--Explanation of resonance.]

Important Topics

1. Musical intervals: octave, sixth, fifth, fourth, third.

2. Major chord, 4:5:6.

3. Use of sharps and flats. Tempered scale.

4. Resonance, sympathetic vibration, explanation, examples.

Exercises

1. What is a major scale? Why is a major scale said to be built upon three triads?

2. Why are sharps and flats necessary in music?

3. What is the tempered scale and why is it used? What instruments need not use it? Why?

4. Mention two examples of resonance or sympathetic vibration from your own experience out of school.

5. An air column 2 ft. long closed at one end is resonant to what wave length? What number of vibrations will this sound have per second at 25C.?

6. At 24C. What length of air column closed at one end will be resonant to a sound having 27 vibrations a second?

7. A given note has 300 vibrations a second. What will be the number of vibrations of its (a) octave, (b) fifth, (c) sixth, (d) major third?

8. In the violin or guitar what takes the place of the sounding board of the piano?

9. Can you explain why the pitch of the bell on a locomotive rises as you rapidly approach it and falls as you recede from it?

10. Do notes of high or low pitch travel faster? Explain.

11. An "A" tuning fork on the "international" scale makes 435 vibrations per second. What is the length of the sound waves produced?

(5) WAVE INTERFERENCE, BEATS, VIBRATION OF STRINGS

=340. Interference of waves.=--The possibility of two trains of waves combining so as to produce a reduced motion or a _complete destruction_ of motion may be shown graphically. Suppose two trains of waves of equal wave length and amplitude as in Fig. 329 meet in _opposite phases_. That is, the parts corresponding to the _crests_ of _A_ coincide with the _troughs_ of _B_, also the troughs of _A_ with the crests of _B_; when this condition obtains, the result is that shown at _C_, the union of the two waves resulting in complete destruction of motion. _The more or less complete destruction of one train of waves by another similar train is an ill.u.s.tration of_ =interference=. If two sets of water waves so unite as to entirely destroy each other the result is a level water surface. If two trains of sound waves combine they may so interfere that silence results. The conditions for securing interference of sound waves may readily be secured by using a tuning fork and a resonating air column. If the tuning fork is set vibrating and placed over the open end of the resonating air column (see Fig. 328), an increase in the sound through resonance may be heard. If the fork is rotated about its axis, in some positions no sound is heard while in other positions the sound is strongly reinforced. Similar effects may be perceived by holding a vibrating fork near the ear and slowly rotating as before. In some positions interference results while in other positions the sound is plainly heard. The explanation of interference may be made clear by the use of a diagram. (See Fig. 330.) Let us imagine that we are looking at the two square ends of a tuning fork. When the fork is vibrating the two p.r.o.ngs approach each other and then recede. As they approach, a condensation is produced at 2 and rarefactions at 1 and 3. As they separate, a rarefaction is produced at 2 and condensations at 1 and 3.

Now along the lines at which the simultaneously produced rarefactions and condensations meet there is more or less complete interference. (See Fig. 331.) These positions have been indicated by dotted lines extending through the ends of the p.r.o.ngs. As indicated above, these positions may be easily found by rotating a vibrating fork over a resonant air column, or near the ear.

[Ill.u.s.tration: FIG. 329.--Interference of sound waves.]

[Ill.u.s.tration: FIG. 330.--At 2 is a condensation; at 1 and 3 are rarefactions.]

[Ill.u.s.tration: FIG. 331.--The condensations and rarefactions meet along the dotted lines producing silence.]

[Ill.u.s.tration: FIG. 332.--Diagram ill.u.s.trating the formations of beats.]

=341. Beats.=--If two tuning forks of slightly different pitch are set vibrating and placed over resonating air columns or with the stem of each fork upon a sounding board, so that the sounds may be intensified, a peculiar pulsation of the sound may be noticed. This phenomenon is known as _beats_. Its production may be easily understood by considering a diagram (Fig. 332). Let the curve _A_ represent the sound wave sent out by one tuning fork and _B_, that sent out by the other. _C_ represents the effect produced by the combination of these waves. At _R_ the two sound waves meet in the _same phase_ and reinforce each other.

This results in a louder sound than either produces alone. Now since the sounds are of slightly different pitch, one fork sends out a few more vibrations per second than the other. The waves from the first fork are therefore a little shorter than those from the other. Consequently, although the two waves are at one time in the _same_ phase, they must soon be in opposite phases as at _I_. Here interference occurs, and silence results. Immediately the waves reinforce, producing a louder sound and so on alternately. The resulting rise and fall of the sound are known as _beats_. The number of beats per second must, of course, be the same as the difference between the numbers of vibrations per second of the two sounds. One effect of beats is _discord_. This is especially noticeable when the number of beats per second is between 30 and 120.

Strike the two lowest notes on a piano at the same time. The beats are very noticeable.

[Ill.u.s.tration: FIG. 333.--Turkish cymbals.]

[Ill.u.s.tration: FIG. 334.--The cornet.]

=342. Three Cla.s.ses of Musical Instruments.=--There are three cla.s.ses or groups of musical instruments, if we consider the vibrating body that produces the sound in each: (A) Those in which the sound is produced by a vibrating _plate_ or _membrane_, as the drum, cymbals (Fig. 333), etc.; (B) those with vibrating _air columns_, as the flute, pipe organ, and cornet (Fig. 334), and (C) with vibrating _wires_ or _strings_, as the piano, violin, and guitar. It is worth while to consider some of these carefully. We will begin with a consideration of vibrating wires and strings, these often producing tones of rich quality.

Let us consider the strings of a piano. (If possible, look at the strings in some instrument.) The range of the piano is 7-1/3 octaves.

Its lowest note, A_{4}, has about 27 vibrations per second. Its highest, C4, about 4176. This great range in vibration rate is secured by varying the length, the tension, and the diameter of the strings.