Chapter III: Part 3

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 pronounced 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
compass 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 mass 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 institution 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 bass 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 bass last and your work is finished. If your first efforts are at all satisfactory you should be greatly encouraged and feel assured 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?

6. With what results have you demonstrated the experiments in this
and the previous lesson?

LESSON XI.

~THE TECHNIQUE OR MODUS OPERANDI OF PIANO TUNING.~

At this juncture, it is thought prudent to defer the discussion of scale building and detail some of the requirements connected with the technical operations of tuning. We do this here because some students are, at this stage, beginning to tune and unless instructed in these things will take hold of the work in an unfavorable way and, perhaps, form habits that will be hard to break. Especially is this so in the matter of setting the mutes or wedges. As to our discussion of scale building, we shall take that up again, that you may be more thoroughly informed on that subject.

Some mechanics do more work in a given time than others, do it as well or better, and with less exertion. This is because they have method or system in their work so that there are no movements lost. Every motion is made to count for the advancement of the cause. Others go about things in a reckless way, taking no thought as to time and labor-saving methods.

In spite of any instruction that can be given, the beginner in piano tuning will not be able to take hold of his work with the ease and the grace of the veteran, nor will he ever be able to work with great accuracy and expedition unless he has a systematic method of doing the various things incident to his profession.

In this lesson, as its subject implies, we endeavor to tell you just how to begin and the way to proceed, step by step, through the work, to obtain the best results in the shortest time, with the greatest ease and the least confusion.

MANIPULATION OF THE TUNING HAMMER.

It may seem that the tightening of a string by turning a pin, around which it is wound, by the aid of an instrument fitting its square end, is such a simple operation that it should require no skill. Simply tightening a string in this manner is, to be sure, a simple matter; but there is a definite degree of tension at which the _vibrating section_ of the string must be left, and it should be left in such a condition that the tension will remain invariable, or as near so as is possible. The only means given the tuner by which he is to bring about this condition are his tuning hammer and the key of the piano, with its mechanism, whereby he may strike the string he is tuning.

The purpose of the tuning hammer is that of altering the tension. The purpose of striking the string by means of the key is twofold: first, to ascertain the pitch of the string, and second, to equalize the tension of the string over its entire length. Consider the string in its three sections, viz.: lower dead end (from hitch pin to lower bridge), vibrating section (section between the bridges), and upper dead end (from upper bridge to tuning pin).

When placing the hammer on the tuning pin and turning to the right, it is evident that the increased tension will be manifest first in the upper dead end. In pianos having agraffes or upper bridges with a tightly screwed bearing bar which makes the strings draw very hard through the bridge, some considerable tension may be produced in the upper dead end before the string will draw through the bridge and increase the tension in the vibrating middle. In other pianos the strings "render" very easily over the upper bridge, and the slightest turn of the hammer manifests an alteration of pitch in the vibrating section. As a rule, strings "render" much more easily through the upper, than the lower bridge. There are two reasons for this: One is, that the construction of the lower bridge is such as to cause a tendency in this direction, having two bridge-pins which stand out of line with the string and bear against it in opposite directions; the other is that the lower bridge is so much farther from the point where the hammer strikes the string that its vibration does not help it through as it does at the upper bridge.

Now, the thing desired is to have the tension equally distributed over the entire length of the string. Tension should be the same in the three different sections. This is of paramount importance. If this condition does not obtain, the piano will not stand in tune. Yet, this is not the only item of importance. The tuning pin must be properly "set," as tuners term it.

By "setting the pins," we mean, leaving it so balanced with respect to the pull of the string that it will neither yield to the pull of the string nor tend to draw it tighter. Coming now to the exact manipulation of the tuning hammer, there are some important items to consider.[F] Now, if the tuning hammer is placed upon the tuning pin with the handle straight upward, and it is pulled backward (from the tuner) just a little, before it is turned to the right, the tension will be increased somewhat before the pin is turned, as this motion, slight as it may seem, pulls the pin upward enough to draw the string through the upper bridge an infinitesimally small distance, but enough to be perceptible to the ear. Now if the hammer were removed, the tendency of the pin would be to yield to the pull of the string; but if the pin is turned enough to take up such amount of string as was pulled through the bridge, and, as it is turned, is allowed to yield downward toward the pull of the string, it will resume its balance and the string will stand at that pitch, provided it has been "rendered" properly over the bridges.

[F] Bear in mind, the foregoing and following instructions are
written with reference to the upright piano. The square does not
permit the observance of these suggestions so favorably as the
upright.

We set forth these details that you may have a thorough understanding of what is meant by setting the pins, and while it is not always advisable to follow this method in tuning, there are some pianos that will stand more satisfactorily when treated in this way. This method is recommended where the string has become rusty at the upper bridge, as it is loosened at the bridge before it is started to wind around the pin which prevents it breaking at that point. We believe that ninety per cent. of strings break right where they start around tuning pin. A very good way to draw a string up is to give the hammer an alternate up and down motion, pulling the handle lightly to you, then from you, as you draw it up; not enough to bend or break the pin or to crush the wood around the pin, but just enough to make the string take on its increased tension equally.

In regard to the lower bridge, the strings will rarely "render" through them properly unless brought to a tension a little higher than it is desired they shall be left. If this is done, a few sharp blows of key will generally make them equalize all right; then press the hammer gently to the left, not enough to turn the pin in the socket, but to settle it back to a well-balanced position. After a little practice the tuner can generally guess precisely how much over-tension to allow. If the pin is left slightly sprung downward, its tendency will be to spring upward, thereby sharpening the string; so be careful to leave the pins in perfect balance, or as tuners say, "properly set."

The foregoing, while applicable to the whole scale, is not so urgent in the over-strung bass. The strings are so heavy and the tension is so great that they will generally "render" quite freely over the bridges, and it is only necessary to bring them up to pitch, handling the hammer in such a manner as to leave the pins well balanced; but it is not necessary to give them over-tension and beat them down again; in fact it is not advisable, as a rule. At all times, place the hammer on the pin as far as it will go, and strike the key while drawing a string up.

In tuning the square piano, it is not possible to set the hammer upon the pin with the handle in line with, and beyond the string, as is the rule in the upright. Where the square has the square pin, the hammer (with star head) can always be set with the handle to the right of the string somewhat, but usually almost in line with the string and almost directly over it, and the manipulation of the hammer is much the same, though the tuner is at a greater disadvantage, the pins being farther from him and he has not such a good rest for his hand. Many old squares have the oblong pin. In this case, use the double hammer head. On the one side the hole in the head is made with the longer diameter in line with the handle, and on the other side the hole is made with the longer diameter at right angles with the handle; so that if you cannot get a favorable position with one end you can with the other.

We have said nothing about which hand to use in striking the keys and in wielding the hammer, but it is customary to handle the hammer with the right hand and it is always advisable for two very good reasons: It gives the tuner a much more favorable position at the instrument; and, as the right hand is more used in ordinary every-day operations and is more trained in applying degrees of force and guiding tools, it is more easily trained to manipulate the hammer properly. Training the hand in the skilful use of the hammer is of the utmost importance and comes only by continued practice, but when it is trained, one can virtually "feel" the tones with the hammer.

At first, the young tuner is almost invariably discouraged by his slow progress. He must remember that, however fine his ear and however great his mechanical ability, he has much to acquire by training in both, and he must expect to be two or three times longer in finishing off a job of tuning at the outset than will be necessary after he has had a few months' practice. You can be your own trainer in these things if you will do a little rational thinking and be content to "hasten slowly." And as to using the left hand, we would not advise it in any event.

SETTING THE MUTES OR WEDGES IN THE UPRIGHT.

As stated in a previous lesson, the mutes should be so placed that only two strings are heard at one time: the one the tuner is tuning, and the one he is tuning by. It is true that this is an easy matter, but it is also true that very few tuners know how to do it in a way to save time and avoid placing the mutes two or more times in the same place. By using a little inventive genius during early practice the author succeeded in formulating a system of muting by which he accomplished the ends as stated above, and assures the reader that a great deal of time can be saved by following it.

After removing the muffler or any other instrumental attachment which may be in the piano in the way of placing the mutes, the first thing to do is to place the continuous mute so that all the outside strings of the trios are damped. The temperament is then set by tuning the middle strings, of the twenty-five trios comprised in the two-octave temperament as demonstrated in a previous lesson. After satisfying yourself by trials or test that the temperament is true, you then remove the continuous mute and proceed to bring the outside strings in unison with the middle one. Now, your 1C is sometimes found to be the first pair in the over-strung bass, which usually has two strings to a key, while in other pianos, 1C is the first trio in the treble stringing, and in many cases it is the second trio in the treble. For illustration, we will say it is the second in the treble. In speaking of the separate strings of a trio we will number them 1st, 2d, and 3d, from left to right, as in foot-note, page 89, Lesson IX. Setting the mutes in bringing up the unisons in the temperament is exceedingly simple.

The following diagram will, we think, demonstrate clearly the method employed:

Upper row---- o o o o o o o o o o Tim-

Middle row--- o o o o o o o o o o ing

Lower row---- o o o o o o o o o o Pins.
------------------------------------------ Bridge.

* 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * &c
/// /// /// /// /// /// /// /// /// /// Treble stringing.
B C C♯ D D♯ E F F♯ G G♯ &c

The upper row of O's represents the upper row of tuning pins. To these are attached the first string of each unison. To the middle row are attached the second or middle strings, and to the lower row are attached the third strings. The diagonal lines represent the three strings of the unison (trio). The asterisk on the middle one indicates that it has been tuned.

But one mute is used in tuning these unisons. It is inserted between the trios in the order indicated by the figures 1, 2, 3, etc. When inserted in place 1, between unisons B and C, it will mute the first string of C; so the first string of the trio to tune is always the third. Then place your mute in place 2 and tune the first string of C. Then, without moving your mute, bring up third string of C♯, then third string of D and so on. By this method, you tune two strings every time you reset your mute.

When through with the temperament, the next step is usually that of tuning the bass; but while we are in the treble we will proceed to give the method of setting the mutes in the upper treble beyond the temperament. All three strings have yet to be tuned here, and we have to use two mutes. The unisons are tuned in regular succession upward the same as in the example above. The mute that is kept farthest to the left, is indicated by the letter A, and the one kept to the right, by the letter B, as in diagram below.

(T e m p e r a m e n t ) 1 2 3 4 5 6 7 &c. *** *** *** *** *** *** A B /// /// /// /// /// /// /// /// /// /// /// /// /// /// C♯ D D♯ E F F♯ G &c.

The mutes are first placed in the places indicated by the figures 1 and 2, thereby muting first and third strings of the first unison beyond the temperament, which is 3C♯. The middle string of this unison is now tuned by its octave below. (If you have left imperfect unisons in your temperament, rendering it difficult to tune octaves by them, it will be well to replace your continuous mute so as to tune from a single string.) Having tuned the middle string of C♯, move mute B to place 3 and tune third string of C♯. Then, move mute A to place 2 and tune first string of C♯. Your mutes are now already set for tuning the middle string of D. After this is done, proceed to move mute B first, then mute A; tuning middle string, then third, then first, moving step by step as indicated in example above until the last unison is reached. By this system you tune three strings every time the mutes are set twice.

The over-strung bass usually has but two strings to a unison and only one mute is needed. In the extreme low or contra-bass, pianos have but one string, in tuning which the mute is discarded. Set the mute as indicated by the figures 1, 2, 3, etc., in the diagram below, always tuning the string farthest to the right by its octave above; then move the mute to its next place and tune the left string by the right. Here, again, you tune two strings every time you reset your mute. The I's represent bass strings.

9 8 7 6 5 4 3 2 1
I I I I I I I II II II II II II II II II
C C♯ D D♯ E F F♯ G G♯ A A♯ B C
Contra-Bass. Bass. Treble.

SETTING THE MUTES IN THE SQUARE PIANO.

In setting the temperament in the square piano, simply mute the string farthest to the left and tune the one to the right until the temperament is finished, then set the mutes in the bass the same as in the upright. In tuning the treble, if the piano has three strings, the same system is used as has been described for the upright. When the piano has but two strings to a unison, as is usually the case, employ the system described for the bass of the upright, but reversed, as you are proceeding to the right instead of to the left.

Remove the shade before beginning to tune a square piano, and if necessary, lay the dampers back and trace the strings to their pins so as to mark them. Certain pins are marked to guide the tuner in placing his hammer. The way we have always marked them is as follows:

Mark both pins of each pair of C strings with white crayon. Mark only one pin of each pair of G's. Knowing the intervals of the other keys from the marked ones, you can easily calculate correctly, upon which pin to set your hammer to tune any string desired. For instance, if you are striking D♯, next above middle C, you calculate that, as D♯ is the third chromatic interval from middle C, you are to set the hammer on one or the other of the pins belonging to the third pair to the right of the pair marked as middle C. B would be first pair to the left, F♯ would be first pair to the left of the marked G, and so on. It is usually necessary to mark only those pairs near the middle of the piano, but we advise the beginner to mark throughout the scale, as by so doing he may avoid breaking a string occasionally by pulling on some other than the one he is sounding. This will occur in your early practice if you do not use caution. And for safety, some tuners always mark throughout.

QUESTIONS ON LESSON XI.

1. By what means is the tuner enabled to make the strings draw
through the bridges and equalize the tension throughout their
entire length?

2. State conditions that may result from a tuning pin's not being
properly set.

3. In this system of muting, state definitely which string is
tuned first after the continuous mute is removed. Which second?
Which third?

4. After the unisons are finished in the temperament, which string
is tuned next, if we go immediately from the temperament to the
over-strung bass? Which second? Which third?

5. Upon beginning to tune the treble beyond the temperament, which
string is tuned first? Which second? Which third?

6. (a) How many mutes are used in tuning outside the strings of
the temperament?

(b) In what proportion is the number of times the mute is
changed to the number of strings tuned?

7. (a) How many mutes are used in tuning the treble beyond the
temperament?

(b) In what proportion is the number of times the mute is
changed to the number of strings tuned?

8. Which pairs of pins are marked in the square piano to guide the
tuner in placing his hammer? Also, how are they marked?

9. Having marked your pins as instructed, how would you find the
pins belonging to a pair of strings struck by F on key-board? How
those struck by G♯?

10. Tell what you can of the requirements necessary to insure that
a piano will stand in tune.

LESSON XII.

~MATHEMATICS OF THE TEMPERED SCALE.~

One of the first questions that arises in the mind of the thinking young tuner is: Why is it necessary to temper certain intervals in tuning? We cannot answer this question in a few words; but you have seen, if you have tried the experiments laid down in previous lessons, that such deviation is inevitable. You know that practical scale making will permit but two pure intervals (unison and octave), but you have yet to learn the scientific reasons why this is so. To do this, requires a little mathematical reasoning.

In this lesson we shall demonstrate the principles of this complex subject in a clear and comprehensive way, and if you will study it carefully you may master it thoroughly, which will place you in possession of a knowledge of the art of which few tuners of the present can boast.

In the following demonstrations of relative pitch numbers, we adopt a pitch in which middle C has 256 vibrations per second. This is not a pitch which is used in actual practice, as it is even below international (middle C 258.65); but is chosen on account of the fact that the various relative pitch numbers work out more favorably, and hence, it is called the "Philosophical Standard." Below are the actual vibration numbers of the two pitches in vogue; so you can see that neither of these pitches would be so favorable to deal with mathematically.

International--3C--517.3. Concert--3C--540.

(Let us state here that the difference in these pitches is less than a half-step, but is so near that it is generally spoken of as being just a half-step.)

Temperament denotes the arrangement of a system of musical sounds in which _each one_ will form a serviceable interval with _any one_ of the others. Any given tone must do duty as the initial or key-note of a major or of a minor scale and also as any other member; thus:

C must serve as 1, in the key of C major or C minor.
" " 2, " " B♭ " B♭ "
" " 3, " " A♭ " A "
" " 4, " " G " G "
" " 5, " " F " F "
" " 6, " " E♭ " E "
" " 7, " " D♭ " C♯ "

Likewise, all other tones of the instrument must be so stationed that they can serve as _any member_ of _any scale_, major or minor.

This is rendered necessary on account of the various modulations employed in modern music, in which every possible harmony in every key is used.

RATIONALE OF THE TEMPERAMENT.

Writers upon the mathematics of sound tell us, experience teaches us, and in previous lessons we have demonstrated in various ways, that if we tune all fifths perfect up to the seventh step (see diagram, pages 82, 83) the last E obtained will be too sharp to form a major third to C. In fact, the third thus obtained is so sharp as to render it offensive to the ear, and therefore unfit for use in harmony, where this interval plays so conspicuous a part. To remedy this, it becomes necessary to tune each of the fifths a very small degree flatter than perfect. The E thus obtained will not be so sharp as to be offensive to the ear; yet, if the fifth be properly altered or tempered, the third will still be sharper than perfect; for if the fifths were flattened enough to render the thirds perfect, they (the fifths) would become offensive. Now, it is a fact, that the third will bear greater deviation from perfect consonance than the fifth; so the compromise is made somewhat in favor of the fifth. If we should continue the series of perfect fifths, we will find the same defect in all the major thirds throughout the scale.

We must, therefore, flatten each fifth of the complete circle, C-G-D-A-E-B-F♯-C♯-G♯ or A♭-E♭-B♭-F-C, successively in a very small degree; the depression, while it will not materially impair the consonant quality of the fifths, will produce a series of somewhat sharp, though still agreeable and harmonious major thirds.

We wish, now, to demonstrate the cause of the foregoing by mathematical calculation, which, while it is somewhat lengthy and tedious, is not difficult if followed progressively. First, we will consider tone relationship in connection with relative string length. Students who have small stringed instruments, guitar, violin, or mandolin, may find pleasure in demonstrating some of the following facts thereupon.

One-half of any string will produce a tone exactly an octave above that yielded by its entire length. Harmonic tones on the violin are made by touching the string lightly with the finger at such points as will cause the string to vibrate in segments; thus if touched exactly in the middle it will produce a harmonic tone an octave above that of the whole string.

Two-thirds of the length of a string when stopped produces a tone a fifth higher than that of the entire string; one-third of the length of a string on the violin, either from the nut or from the bridge, if touched lightly with the finger at that point, produces a harmonic tone an octave higher than the fifth to the open tone of that string, because you divide the string into three vibrating segments, each of which is one-third its entire length. Reason it thus: If two-thirds of a string produce a fifth, one-third, being just half of two-thirds, will produce a tone an octave higher than two-thirds. For illustration, if the string be tuned to 1C, the harmonic tone produced as above will be 2G. We might go on for pages concerning harmonics, but for our present use it is only necessary to show the general principles. For our needs we will discuss the relative length of string necessary to produce the various tones of the diatonic scale, showing ratios of the intervals in the same.

In the following table, 1 represents the entire length of a string sounding the tone C. The other tones of the ascending major scale require strings of such fractional length as are indicated by the fractions beneath them. By taking accurate measurements you can demonstrate these figures upon any small stringed instrument.

Funda- | Major | Major | Perfect | Perfect | Major | Major | Oc- |
mental |Second | Third | Fourth | Fifth | Sixth | Seventh | tave |
| | | | | | | |
C | D | E | F | G | A | B | C |
1 | 8/9 | 4/5 | 3/4 | 2/3 | 3/5 | 8/15 | 1/2 |

To illustrate this principle further and make it very clear, let us suppose that the entire length of the string sounding the fundamental C is 360 inches; then the segments of this string necessary to produce the other tones of the ascending major scale will be, in inches, as follows:

C | D | E | F | G | A | B | C | 360 | 320 | 288 | 270 | 240 | 216 | 192 | 180 |

Comparing now one with another (by means of the ratios expressed by their corresponding numbers) the intervals formed by the tones of the above scale, it will be found that they all preserve their original purity except the minor third, D-F, and the fifth, D-A. The third, D-F, presents itself in the ratio of 320 to 270 instead of 324 to 270 (which latter is equivalent to the ratio of 6 to 5, the true ratio of the minor third). The third, D-F, therefore, is to the true minor third as 320 to 324 (reduced to their lowest terms by dividing both numbers by 4, gives the ratio of 80 to 81). Again, the fifth, A-F, presents itself in the ratio of 320 to 216, or (dividing each term by 4) 80 to 54; instead of 3 to 2 (=81 to 54--multiplying each term by 27), which is the ratio of the true fifth. Continuing the scale an octave higher, it will be found that the sixth, F-D, and the fourth, A-D, will labor under the same imperfections.

The comparison, then, of these ratios of the minor third, D-F, and the fifth, D-A, with the perfect ratios of these intervals, shows that each is too small by the ratio expressed by the figures 80 to 81. This is called, by mathematicians, the _syntonic comma_.

As experience teaches us that the ear cannot endure such deviation as a whole comma in any fifth, it is easy to see that some tempering must take place even in such a simple and limited number of sounds as the above series of eight tones.

The necessity of temperament becomes still more apparent when it is proposed to combine every sound used in music into a connected system, such that each individual sound shall not only form practical intervals with all the other sounds, but also that each sound may be employed as the root of its own major or minor key; and that all the tones necessary to form its scale shall stand in such relation to each other as to satisfy the ear.

The chief requisites of any system of musical temperament adapted to the purposes of modern music are:--

1. That all octaves must remain perfect, each being divided into
twelve semitones.

2. That each sound of the system may be employed as the root of a
major or minor scale, without increasing the number of sounds in
the system.

3. That each consonant interval, according to its degree of
consonance, shall lose as little of its original purity as
possible; so that the ear may still acknowledge it as a perfect or
imperfect consonance.

Several ways of adjusting such a system of temperament have been proposed, all of which may be classed under either the head of equal or of unequal temperament.

The principles set forth in the following propositions clearly demonstrate the reasons for tempering, and the whole rationale of the system of equal temperament, which is that in general use, and which is invariably sought and practiced by tuners of the present.

PROPOSITION I.

If we divide an octave, as from middle C to 3C, into three major thirds, each in the perfect ratio of 5 to 4, as C-E, E-G♯ (A♭), A♭-C, then the C obtained from the last third, A♭-C, will be too flat to form a perfect octave by a small quantity, called in the theory of harmonics a _diesis_, which is expressed by the ratio 128 to 125.

EXPLANATION.--The length of the string sounding the tone C is represented by unity or 1. Now, as we have shown, the major third to that C, which is E, is produced by 4/5 of its length.

In like manner, G♯, the major third to E, will be produced by 4/5 of that segment of the string which sounds the tone E; that is, G♯ will be produced by 4/5 of 4/5 (4/5 multiplied by 4/5) which equals 16/25 of the entire length of the string sounding the tone C.

We come, now, to the last third, G♯ (A♭) to C, which completes the interval of the octave, middle C to 3C. This last C, being the major third from the A♭, will be produced as before, by 4/5 of that segment of the string which sounds A♭; that is, by 4/5 of 16/25, which equals 64/125 of the entire length of the string. Keep this last fraction, 64/125, in mind, and remember it as representing the segment of the entire string, which produces the upper C by the succession of three perfectly tuned major thirds.

Now, let us refer to the law which says that a perfect octave is obtained from the exact half of the length of any string. Is 64/125 an exact half? No; using the same numerator, an exact half would be 64/128.

Hence, it is clear that the octave obtained by the succession of perfect major thirds will differ from the true octave by the ratio of 128 to 125. The fraction, 64/125, representing a longer segment of the string than 64/128 (1/2), it would produce a flatter tone than the exact half.

It is evident, therefore, that _all major thirds must be tuned somewhat sharper than perfect_ in a system of equal temperament.

The ratio which expresses the value of the _diesis_ is that of 128 to 125. If, therefore, the octaves are to remain perfect, which they must do, _each major third must be tuned sharper than perfect by one-third part of the diesis_.

The foregoing demonstration may be made still clearer by the following diagram which represents the length of string necessary to produce these tones. (This diagram is exact in the various proportional lengths, being about one twenty-fifth the actual length represented.)

Middle C (2C) 60 inches.
--------------------------------------------------
O O

E (4/5 of 60) 48 inches.
--------------------------------------------
O O

G♯ (A♭) (4/5 of 48) 38-2/5 inches.
--------------------------------------
O O

3C (4/5 of 38-2/5) 30-18/25 inches.
--------------------------------
O O

This diagram clearly demonstrates that the last C obtained by the succession of thirds covers a segment of the string which is 18/25 longer than an exact half; nearly three-fourths of an inch too long, 30 inches being the exact half.

To make this proposition still better understood, we give the comparison of the actual vibration numbers as follows:--

Perfect thirds in ratio 4/5 have these vibration numbers: =

1st third 2d third 3d third
(C 256 - E 320) (E 320 - G♯ 400) (G♯ 400 - C 500)
--------------- ----------------- -----------------
no beats no beats no beats

Tempered thirds qualified to produce true octave: =

(C 256 - E 322 5/10) (E 322 5/10 - G♯ 406 4/10) (G♯ 406 4/10 - C 512)
-------------------- -------------------------- ----------------------
10 beats 13-1/10 beats 16 beats

We think the foregoing elucidation of Proposition I sufficient to establish a thorough understanding of the facts set forth therein, if they are studied over carefully a few times. If everything is not clear at the first reading, go over it several times, as this matter is of value to you.

QUESTIONS ON LESSON XII.

1. Why is the pitch, C-256, adopted for scientific discussion, and
what is this pitch called?

2. The tone G forms the root (1) in the key of G. What does it
form in the key of C? What in F? What in D?

3. What tone is produced by a 2/3 segment of a string? What by a
1/2 segment? What by a 4/5 segment?

4. (a) What intervals must be tuned absolutely perfect?

(b) In the two intervals that must be tempered, the third and the
fifth, which will bear the greater deviation?

5. What would be the result if we should tune from 2C to 3C by a
succession of perfect thirds?

6. Do you understand the facts set forth in Proposition I, in this
lesson?

LESSON XIII.

~RATIONALE OF THE TEMPERAMENT.~ (Concluded from Lesson XII.)

PROPOSITION II.

That the student of scientific scale building may understand fully the reasons why the tempered scale is at constant variance with exact mathematical ratios, we continue this discussion through two more propositions, No. II, following, demonstrating the result of dividing the octave into four minor thirds, and Proposition III, demonstrating the result of twelve perfect fifths. The matter in Lesson XII, if properly mastered, has given a thorough insight into the principal features of the subject in question; so the following demonstration will be made as brief as possible, consistent with clearness.