Lesson VIII.

The Temperament.

Some tuners favor the term, “laying the bearings,” others say “setting the temperament.” The former is more commonplace, as it merely suggests the idea of laying a number of patterns by which all others are to be measured. The latter term is extremely comprehensive. A lucid definition of the word “temperament,” in the sense in which it is used here, would require a discourse of considerable length. The following statements will elicit the full meaning of the term:

The untutored would, perhaps, not think of setting a temperament to tune by. He would likely begin at some unfavorable point, and tune by various intervals, relying wholly upon his conception of pitch for the accuracy of the tones tuned, the same as a violinist in tuning his four strings. To be sure, pitch has to be reckoned as a rude guide in setting the tones; but if pitch alone were the guide we would never attain to any degree of perfection in scale forming. We could never adjust our tones to that delicate fineness so much appreciated, which gives to the instrument its surpassing brilliancy.

Beats, Waves, Pulsations.—To obtain absolute accuracy the tuner is guided by beats, waves or pulsations. These three words refer to one and the same thing, a phenomenon that occurs in certain intervals when two tones are sounded together that are not in exact tune. These terms must not be confounded with the term “sound wave” or “vibrations” so often used in discussions on the theory of sound. However, we think the student is thoroughly familiar with these terms. The rate of vibration of two tones not in a favorable ratio, may produce the phenomenon known as “beats, waves, or pulsations.” Vibrations may exist either with or without pulsations.

These pulsations are most perceptible in the unison, the octave and the fifth. They are more easily perceived in the unison than in the octave, and more easily in the octave than in the fifth. They are also perceptible in the perfect fourth, the major and minor third and some other intervals, but on account of their obscurity, and because these intervals are unnecessary in tuning they have long since been abandoned in “temperament making” (with the exception of the perfect fourth) by most tuners, although a few still make use of it. We do not say that the fourth is unsafe to tune by, but you will see later on why it is not best to make use of it.

The Fischer System or method of “setting the temperament” has these advantages: It uses but two kinds of intervals: the fifth and octave; by employing two whole octaves in place of one or one and a half, nearly all of the middle section of strings is brought up in pitch which insures that the temperament will stand better while the remaining strings are being tuned; and the alternate tuning of the fifth and octave makes the system exceedingly easy to learn, enabling the tuner to work with less mental strain. Also the two-octave system gives a greater compass for testing, thus insuring greater accuracy.

If you have access to a piano, it will now be well for you to begin training the ear to perceive the pulsations. If you cannot use a piano, you can train very well by the use of a mandolin, guitar, violin, zither, or any stringed instrument. An instrument with metal strings, however, is better, as the vibrations are more perfect.

You will, of course, know that the front top panel of the case has to be removed to give access to the tuning pins, and that you should have a regular tuning hammer and set of mutes to begin with. The panel is held in place in various ways: sometimes with buttons, sometimes with pins set in slots, and sometimes with patent fastenings; but a little examination will reveal how it may be removed.

To produce a tone of a certain pitch, the string must be of the right thickness and length. These items are decided by the scale draughtsman in the factory; if incorrect, the tuner can do nothing to improve them.

To produce the correct pitch, the string must be of the right tension, which is brought about by winding one end of the string around the tuning pin until the proper degree of tension is reached. This must be decided by the ear of the tuner. Two strings of equal thickness and equal length produce the same tone when brought to the same tension; the result being known as “unison.” A defect in the unison being the easiest way in which to detect the beats, we advise that the student practice on it first.

After taking out the panel, the first thing to do is to place your rubber mute between two trios of strings (if the piano is an upright which usually has three strings to a note) so that only two strings sound when the key is struck. Select some key near the middle of the keyboard. Strike the key strongly and hold it down. If the two sounding strings give forth a smooth, unwavering tone—a tone that sounds as if it came from one string, the unison is perfect. If you find it so, remove the mute and place it on the other side of the trio of strings. If the piano has been tuned recently by an expert, you may have to continue your search over several keys before you find an imperfect unison; but you will rarely find a piano in such perfect tune that it will not contain some defective unisons. However, if you do not succeed in finding a defective unison, select a key near the middle of the key-board, place your mute so that but two strings sound, and with your tuning hammer loosen one of the strings very slightly. Now you will notice a throbbing, beating sound, very unlike the tone produced when the strings were in exact unison. See it you can count the beats. If you have lowered the tension too much, the beats will be too rapid to permit counting. Now with a steady and gradual pull, with the heel of the hand against some stationary part, bring the string up slowly. You will notice these waves become slower and slower. When they become quite slow, stop and count, or wave the hand in time with the pulsations. After practicing this until you are sure your ear has become accustomed to the beats and will recognize them again, you may proceed to perfect the unison. Bring the string up gradually as before, and when the unison is reached you will hear one single, simple, musical tone, as though it were from a single string. Never have more than two strings sounding at once. You might go over the entire key-board now and correct all the unisons if the scale is yet fairly good. See which string is, in your opinion, the nearest to correctness with respect to the scale, and tune the other one, or two, as the case may be, to it. If the scale is badly out of symmetry, you will not get very good results without setting a temperament; but the tones will sound better individually. This experiment is more for practice than for improving the piano.

The cause of the waves in a defective unison is the alternate recurring of the periods when the condensations and rarefactions correspond in the two strings and then antagonize. This is known in physics as “interference of sound-waves.”

The Octave.—When perfectly tuned, the upper tone of the octave has exactly double the number of vibrations of the lower. If the lower tone vibrates 1000 per second, the upper will vibrate 2000. Of course, the ear cannot ascertain in any way the number of vibrations per second; we use these figures for scientific demonstration only. However, there is an instrument called the Siren which is constructed for the purpose of ascertaining the number of vibrations per second of any given tone, and which is delicately accurate in its work. By its assistance we know, definitely, a great many things regarding our musical scale of which we would otherwise be ignorant. But, while we cannot, by the ear, ascertain these numbers, we can, by the “interference of sound-waves” above referred to, ascertain, to the most delicate point, when the relative vibration of two strings is mathematically exact, if they are tuned to a unison, octave, fifth, etc.

Practice now on tuning the octave. Find an octave in which the upper tone is flat. Mute all but one string in the lower tone to make sure of getting a pure tone, then select one string (the middle one if a piano has three strings) of the upper octave and proceed to pull it up gradually until all beats disappear. This being done, bring up the unisons.

The Fifth.—In our system, when we speak of a fifth, we mean a fifth upward. The fifth to C is G, to G is D, and so on.

The vibration of the fifth is one and a half times that of its fundamental. If a certain F vibrates 100, the C, a fifth above, will vibrate 150, if tuned so that no waves are heard; but for reasons which will be fully explained later, the fifth cannot be tuned with mathematical precision. On account of certain peculiarities in our tempered scale, the fifth must always be left somewhat flatter than perfect. This fact is always learned with some astonishment by beginners.

In your practice on tuning the fifth, first tune it perfectly, so that no waves are perceptible; then flat it so that there are very slow waves; less than one per second. Some authorities say there should be three beats in five seconds; but the tuner must learn to determine this by his own judgment. The tempering of the fifth will be treated exhaustively in subsequent lessons.

We advise that you confine your practice to the unison until you are sure you have a clear conception of its peculiarities in all portions of the key-board, except the extreme lower and upper octaves; do not try these yet. Do not begin to practice on the octave until you are very familiar with the beats in the unison. By gradual progress you will avoid confusing the ear, each step being thoroughly mastered before advancing to the next. Remember, there is nothing that is extremely difficult in learning to tune if you but understand what has to be done, go about it systematically, and have plenty of patience.

In this lesson we give you our system of setting the temperament; that is, the succession in which the different tones of the temperament are tuned. We advise, however, that you do not attempt to set a temperament until after studying Lesson IX, which enters into the theory of temperament, testing, etc.

Two octaves are used for the temperament: an octave above, and an octave below middle C. Middle C can be told by its being the C nearest the name of the piano on the name board. In other words, it is the fifth C from the highest C, and the fourth from the lowest in the modern piano, which has seven and a third octaves.

The diagram illustrates the two octaves of the key-board, and shows how each key is designated in giving the system of temperament.

Pitch. —The Piano Manufacturers’ Association has established what is known as “international pitch.” Tuning-forks made to this pitch are marked “C-517.3,” meaning that our 3C vibrates 517.3 per second. Concert pitch is nearly a half step higher than this. Some manufacturers still tune their instruments to this higher pitch.

If it is desired to tune a piano to a certain pitch, say concert pitch, tune the C that is an octave above middle C by a concert pitch tuning-fork or pipe. If, however, the piano is too much below that, it is not safe to bring it up to it at one tuning. But, say it will permit tuning to concert pitch; after this C (3C) is well laid, tune middle C (2C) by it, then tune the C octave below middle C (1C) to middle C. Having 1C for a starting point, proceed by tuning a fifth up, then its octave, then a fifth, then an octave, always tuning the octave whichever way is necessary to keep within the two octaves.

The simplicity of this system can be readily seen; yet for the use of beginners, we give on the following page the whole succession of intervals as they are taken in setting the temperament.

Diagram of the Two Octaves Used in “Temperament,” and of the Succession in Which They Are Tuned.

Middle C begins second octave; known by the asterisk (*) under it.

The Fischer System of Setting Temperament.

First, tune 3C by tuning pipe, or as directed.

By this, tune 2C, and by 2C tune 1C; then tune as follows:

You will observe this last fifth brings you back to the starting-point (C). It is called the “wolf,” from the howling of its beats when the tuner has been inaccurate or the piano fails to stand.

Questions on Lesson VIII.

1. What is the cause of the beats or pulsations?
2. Have you practiced tuning the unison?
3. Can you distinguish the beats clearly?
4. Have you practiced tuning the octave?
5. Do you thoroughly understand the system of setting the temperament as set forth in this lesson?