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
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.