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?