[C] The three strings composing the trio or unison are numbered 1st, 2d or middle, and 3d, from left to right.

It is presumed that you are now familiar with the succession of tones and intervals used in setting the temperament. Fix these things in your mind and the system is easy to understand and remember. Keep within the bounds of the two octaves laid out in Lesson X. Tune all fifths upward; that is, tune all fifths by their fundamentals. For example, starting on 1C, use it as fundamental, and by it, tune its fifth, which is G; then, having G tuned, use it as fundamental, and by it tune its fifth, which is D, and so on through. After tuning a fifth, always tune its octave either above or below, whichever way it lies within the bounds of the two octaves. After going through one or two experiments in setting temperament you will see the simplicity of this system and will, perhaps, not be obliged to refer to the diagram any more.

For various reasons, it is better to try your experiments on an upright piano, and the better the piano, the more satisfactory will be the result of the experiment. You should have no hesitancy or timidity in taking hold of a good piano, as you cannot damage it if you use good judgment, follow instructions, and work carefully. The first caution is, be very careful that you draw a string but slightly sharper than it is to be left. Rest the heel of the hand against some stationary part of the piano and pull very slowly, and in a direct right angle with the tuning pin so as to avoid any tendency to bend or spring the pin. We would advise now that you find an upright piano that is badly out of tune, if you have none of your own, and proceed to set a temperament.

The following instructions will suffice for your first experiments, and by them you may be able to get fairly good results; however, the theory of temperament, which is more thoroughly entered into in Lesson XII, must be studied before you can have a thorough understanding of the causes and effects.

After deciding, as per instructions on pitch which C you will tune first, place the tuning hammer (using the star head if pins are square) on the pin with the handle extending upwards or inclined slightly to the right. (The star head, which will fit the pin at eight different angles, enables the tuner to select the most favorable position.) To raise the pitch, you will, of course, pull the hammer to the right. In order to make a string stand in tune, it is well to draw it very slightly above the pitch at which it is to remain, and settle it back by striking the key repeatedly and strongly, and at the same time bearing gently to the left on the tuning hammer. The exact amount of over-tension must be learned by practice; but it should be so slight as to be barely perceptible. Aim to get the string tuned with the least possible turning of the hammer. The tension of the string should be evenly distributed over its entire length; that is, over its vibrating middle and its "dead ends" beyond the bridges. Therefore it is necessary to strike the key strongly while tuning so as to make the string draw through the bridges. By practice, you will gain control of the hammer and become so expert that you can feel the strings draw through the bridges and the pins turn in the block.

Having now tuned your three Cs, you will take 1C as a starting point, and by it, tune 1G a perfect fifth above. Tune it perfect by drawing it gradually up or down until all pulsations disappear. Now after making sure you have it perfect, flatten it until you can hear slow, almost imperceptible waves; less rapid than one per second. This flattening of the fifth is called tempering, and from it comes the word "temperament." The fact that the fifth must always be tuned a little flatter than perfect, is a matter which always causes some astonishment when first learned. It seems, to the uninitiated, that every interval should be made perfect; but it is impossible to make them so, and get a correct scale, as we shall see later on.

Now tune 2G by the 1G just tuned, to a perfect octave. Remember that all octaves should be left perfect--all waves tuned out. Now try 2G with 2C. If your octaves are perfect, this upper fifth will beat a little faster than the lower one, but the dissonance should not be so great as to be disagreeable. Proceed to your next fifth, which is 2D, then its octave, 1D, then its fifth and so on as per directions on the system card. You can make no chord trials until you have tuned E, an interval of a major third from C.

Having tuned 2E, you can now make your first trial: the chord of C. If you have tempered your fifths correctly, this chord will come out in pleasing harmony, and yet the E will be somewhat sharper than a perfect major third to C. Now, just for experiment, lower 2E until all waves disappear when sounded with 2C. You now have a perfect major third. Upon sounding the chord, you will find it more pleasing than before; but you cannot leave your thirds perfect. Draw it up again to its proper temperament with A, and you will notice it has very p.r.o.nounced beats when sounded with C. Proceed with the next step, which is that of tuning 1B, fifth to 1E. When tuned, try it as a major third in the chord of G. At each step from this on, try the note just tuned as a major third in its proper chord. Remember, the third always sounds better if lower than you dare to leave it; but, on the other hand, it must not be left so sharp as to be at all unpleasant when heard in the chord. As to the position of the chord for these trials, the second position, that is, with the third the highest, is the most favorable, as in this position you can more easily discern excessive sharpness of the third, which is the most common occurrence. When you have gone through the entire system and arrived at the last fifth, 1F-2C, you should find it nearly as perfect as the rest, but you will hardly be able to do so in your first efforts. Even old tuners frequently have to go over their work a second or third time before all fifths are properly tempered. By this system, however, you cannot go far wrong if you test each step as directed, and your first chord comes up right. If the first test, G-C-E, proves that there is a false member in the chord, do not proceed with the system, but go over the first seven steps until you find the offending members and rectify.

Do not be discouraged on account of failures. No one ever set a correct temperament at the first attempt.

QUESTIONS ON LESSON IX.

1. Define the terms, "International Pitch," and "Concert Pitch."

2. How would you arrive at the most favorable pitch at which to tune a piano, if the owner did not suggest any certain pitch?

3. What is the advantage in using the continuous mute?

4. Tell what is necessary in the tuning of a string to insure it to stand well?

5. What would result in the major third C-E, if all the fifths, up to E, were tuned perfect?

LESSON X.

~THEORY OF THE TEMPERAMENT.~

The instructions given in Lessons VIII and IX cover the subject of temperament pretty thoroughly in a way, and by them alone, the student might learn to set a temperament satisfactorily; but the student who is ambitious and enthusiastic is not content with a mere knowledge of how to do a thing; he wants to know why he does it; why certain causes produce certain effects; why this and that is necessary, etc. In the following lessons we set forth a comprehensive demonstration of the theory of Temperament, requirements of the correct scale and the essentials of its mathematics.

~Equal Temperament.~--Equal temperament is one in which the twelve fixed tones of the chromatic scale[D] are equidistant. Any chord will be as harmonious in one key as in another.

[D] The chromatic scale is a succession of all the half steps in the compa.s.s of one octave. Counting the octave tone, it contains thirteen tones, but we speak of twelve, as there are only twelve which differ in name.

~Unequal Temperament.~--Unequal temperament was practiced in olden times when music did not wander far from a few keys which were favored in the tuning. You will see, presently, how a temperament could be set in such a way as to favor a certain key (family of tones) and also those keys which are nearly related to it; but, that in favoring these keys, our scale must be constructed greatly to the detriment of the "remote" keys. While a chord or progression of chords would sound extremely harmonious in the favored keys, they would be so unbalanced in the remote keys as to render them extremely unpleasant and almost unfit to be used. In this day, when piano and organ music is written and played in all the keys, the unequal temperament is, of course, out of the question. But, strange to say, it is only within the last half century that the system of equal temperament has been universally adopted, and some tuners, even now, will try to favor the flat keys because they are used more by the ma.s.s of players who play little but popular music, which is mostly written in keys having flats in the signature.

Upon the system table you will notice that the first five tones tuned (not counting the octaves) are C, G, D, A and E; it being necessary to go over these fifths before we can make any tests of the complete major chord or even the major third. Now, just for a proof of what has been said about the necessity of flattening the fifths, try tuning all these fifths perfect. Tune them so that there are absolutely no waves in any of them and you will find that, on trying the chord G-C-E, or the major third C-E, the E will be very much too sharp. Now, let your E down until perfect with C, all waves disappearing. You now have the most perfect, sweetest harmony in the chord of C (G, C, E) that can be produced; all its members being absolutely perfect; not a wave to mar its serene purity. But, now, upon sounding this E with the A below it, you will find it so flat that the dissonance is unbearable. Try the minor chord of A (A-C-E) and you will hear the rasping, throbbing beats of the too greatly flattened fifth.

So, you see, we are confronted with a difficulty. If we tune our fifths perfect (in which case our fourths would also be perfect), our thirds are so sharp that the ear will not tolerate them; and, if we tune our thirds low enough to banish all beats, our fifths are intolerably flat.

The experiment above shows us beautifully the prominent inconsistency of our scale. We have demonstrated, that if we tune the members of the chord of C so as to get absolutely pure harmony, we could not use the chord of A on account of the flat fifth E, which did duty so perfectly as third in the chord of C.

There is but one solution to this problem: Since we cannot tune either the fifth or the third perfect, we must compromise, we must strike the happy medium. So we will proceed by a method that will leave our fifths flatter than perfect, but not so much as to make them at all displeasing, and that will leave our thirds sharper than perfect, but not intolerably so.

We have, thus far, spoken only of the octave, fifth and third. The inquisitive student may, at this juncture, want to know something about the various other intervals, such as the minor third, the major and minor sixth, the diminished seventh, etc. But please bear in mind that there are many peculiarities in the tempered scale, and we are going to have you fully and explicitly informed on every point, if you will be content to absorb as little at a time as you are prepared to receive. While it may seem to us that the tempered scale is a very complex inst.i.tution when viewed as a specific arrangement of tones from which we are to derive all the various kinds of harmony, yet, when we consider that the chromatic scale is simply a series of twelve half-steps--twelve perfectly similar intervals--it seems very simple.

Bear in mind that the two cardinal points of the system of tuning are:

1. All octaves shall be tuned perfect.

2. All fifths shall be tuned a little flatter than perfect.

You have seen from Lesson VIII that by this system we begin upon a certain tone and by a circle of twelve fifths cover every chromatic tone of the scale, and that we are finally brought around to a fifth, landing upon the tone upon which we started.

So you see there is very little to remember. Later on we will speak of the various other intervals used in harmony: not that they form any prominent part in scale forming, for they do not; but for the purpose of giving the learner a thorough understanding of all that pertains to the establishing of a correct equal temperament.

If the instruction thus far is understood and carried out, and the student can properly tune fifths and octaves, the other intervals will take care of themselves, and will take their places gracefully in any harmony in which they are called upon to take part; but if there is a single instance in which an octave or a fifth is allowed to remain untrue or untempered, one or more chords will show it up. It may manifest itself in one chord only. A tone may be untrue to our tempered scale, and yet sound beautifully in certain chords, but there will always be at least one in which it will "howl." For instance, if in the seventh step of our system, we tune E a little too flat, it sounds all the better when used as third in the chord of C, as we have shown in the experiment mentioned on page 94 of this lesson. But, if the remainder of the temperament is accurate, this E, in the chord in which E acts as tonic or fundamental, will be found to be too flat, and its third, G sharp, will demonstrate the fact by sounding too sharp.

The following suggestions will serve you greatly in testing: When a third sounds disagreeably sharp, one or more fifths have not been sufficiently flattened.[E] While it is true that thirds are tuned sharp, there is a limit beyond which we cannot go, and this excessive sharpness of the third is the thing that tuners always listen for.

[E] In making these suggestions, no calculation is made for the liability of the tones tuned to fall. This often happens, in which case your first test will display a sharp third. In cases like this it is best to go on through, taking pains to temper carefully, and go all over the temperament again, giving all the strings an equal chance to fall. If the piano is very bad, you may have to bring up the unisons roughly, inuring this portion of the instrument to the increased tension, when you may again place your continuous mute and set your temperament with more certainty.

The fundamental sounds better to the ear when too sharp. The reason for this is the same as has already been explained above; namely, if the fundamental is too sharp the third will be less sharp to it, and, therefore, nearer perfect.

After you have gone all over your temperament, test every member of the chromatic scale as a fundamental of a chord, as a third, and as a fifth. For instance: try middle C as fundamental in the chord of C (G-C-E or E-G-C or C-E-G). Then try it as third in the chord A flat (E flat-A flat-C or C-E flat-A flat or A flat-C-E flat). Then try it as fifth in the chord of F (C-F-A or A-C-F or F-A-C). Take G likewise and try it as fundamental in the chord of G in its three positions, then try it as a third in the chord of E flat, then as fifth in the chord of C. In like manner try every tone in this way, and if there is a falsely tempered interval in the scale you will be sure to find it.

You now understand that the correctness of your temperament depends entirely upon your ability to judge the degree of flatness of your fifths; provided, of course, that the strings stand as tuned. We have told you something about this, but you may not be able at once to judge with sufficient accuracy to insure a good temperament. Now, we have said, let the fifths beat a little more slowly than once a second; but the question crops up, How am I to judge of a second of time? The fact is that a second of time is quickly learned and more easily estimated, perhaps, than any other interval of time; however, we describe here a little device which will accustom one to estimate it very accurately in a short time. The pendulum oscillates by an invariable law which says that a pendulum of a certain length will vibrate always in a corresponding period of time, whether it swings through a short arc or a long one. A pendulum thirty-nine and a half inches long will vibrate seconds by a single swing; one nine and seven-eighths inches long will vibrate seconds at the double swing, or the to-and-fro swing. You can easily make one by tying any little heavy article to a string of either of these lengths. Measure from the center of such heavy article to the point of contact of the string at the top with some stationary object. This is a sure guide. Set the pendulum swinging and count the vibrations and you will soon become quite infallible. Having acquired the ability to judge a second of time you can go to work with more confidence.

Now, as a matter of fact, in a scale which is equally tempered, no two fifths beat exactly alike, as the lower a fifth, the slower it should beat, and thus the fifths in the ba.s.s are hardly perceptibly flat, while those in the treble beat more rapidly. For example, if a certain fifth beat once a second, the fifth an octave higher will beat twice a second, and one that is two octaves higher will beat four times a second, and so on, doubling the number of beats with each ascending octave.

In a subsequent lesson, in which we give the mathematics of the temperament, these various ratios will be found accurately figured out; but for the present let us notice the difference between the actual tempered scale and the exact mathematical scale in the point of the flattening of the fifth. Take for example 1C, and for convenience of figuring, say it vibrates 128 per second. The relation of a fundamental to its fifth is that of 2 to 3. So if 128 is represented as 2, we think of it as 2 times 64. Then with another 64 added, we have 192, which represents 3. In other words, a fundamental has just two-thirds of the number of vibrations per second that its fifth has, in the exact scale. This would mean a fifth in which there would be no beats. Now in the tempered scale we find that G vibrates 191.78 instead of 192; so we can easily see how much variation from the mathematical standard there is in this portion of the instrument. It is only about a fourth of a vibration. This would mean that, in this fifth we would hear the beats a little slower than one per second.

Take the same fifth an octave higher and take 2C as fundamental, which has 256 for its vibration number. The G, fifth above, should vibrate 384, but in the tempered scale it beats but 383.57, almost half a vibration flat. This would give nearly 2 beats in 3 seconds.

These figures simply represent to the eye the ratios of these sounds, and it is not supposed that a tuner is to attain to such a degree of accuracy, but he should strive to arrive as near it as possible.

It is well for the student to practice temperament setting and regular tuning now if he can do so. After getting a good temperament, proceed to tune by octaves upward, always testing the tone tuned as a fifth and third until his ear becomes sufficiently true on the octave that testing otherwise is unnecessary. Tune the overstrung ba.s.s last and your work is finished. If your first efforts are at all satisfactory you should be greatly encouraged and feel a.s.sured that accuracy will reward continued practice.

QUESTIONS ON LESSON X.

1. What is meant by the term "equal temperament"?

2. What is meant by the term "unequal temperament"?

3. Webster defines the term "temperament" thus: "A system of compromises in the tuning of pianofortes, organs, etc." Explain fully what these compromises are.

4. In testing chords to ascertain if temperament is correct, what is the main thing to listen for as a guide?

5. In what three chords would you try the tone A, in testing your temperament?