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.
Let us figure the result of dividing an octave into four minor thirds.
The ratio of the length of string sounding a fundamental, to the
length necessary to sound its minor third, is that of 6 to 5. In other
words, 5/6 of any string sounds a tone which is an exact minor third
above that of the whole string.
Now, suppose we select, as before, a string sounding middle C, as the
fundamental tone. We now ascend by minor thirds until we reach the C,
octave above middle C, which we call 3C, as follows:
Middle C-E♭; E♭-F♯; F♯-A; A-3C.
Demonstrate by figures as follows:--Let the whole length of string
sounding middle C be represented by unity or 1.
E♭ will be sounded by 5/6 of the string 5/6
F♯, by 5/6 of the E♭ segment; that is, by 5/6 of
5/6 of the entire string, which equals 25/36
A, by 5/6 of 25/36 of entire string, which equals 125/216
3C, by 5/6 of 125/216 of entire string, which equals 625/1296
Now bear in mind, this last fraction, 625/1296, represents the segment
of the entire string which should sound the tone 3C, an exact octave
above middle C. Remember, our law demands an exact half of a string by
which to sound its octave. How much does it vary? Divide the
denominator (1296) by 2 and place the result over it for a numerator,
and this gives 648/1296, which is an exact half. Notice the
comparison.
3C obtained from a succession of exact minor thirds, 625/1296
3C obtained from an exact half of the string 648/1296
Now, the former fraction is smaller than the latter; hence, the
segment of string which it represents will be shorter than the exact
half, and will consequently yield a sharper tone. The denominators
being the same, we have only to find the difference between the
numerators to tell how much too short the former segment is. This
proves the C obtained by the succession of minor thirds to be too
short by 23/1296 of the length of the whole string.
If, therefore, all octaves are to remain perfect, it is evident that
_all minor thirds must be tuned flatter than perfect_ in the system of
equal temperament.
The ratio, then, of 648 to 625 expresses the excess by which the true
octave exceeds four exact minor thirds; consequently, each minor third
must be flatter than perfect by one-fourth part of the difference
between these fractions. By this means the dissonance is evenly
distributed so that it is not noticeable in the various chords, in the
major and minor keys, where this interval is almost invariably
present. (We find no record of writers on the mathematics of sound
giving a name to the above ratio expressing variance, as they have to
others.)
PROPOSITION III.
Proposition III deals with the perfect fifth, showing the result from
a series of twelve perfect fifths employed within the space of an
octave.
METHOD.--Taking 1C as the fundamental, representing it by unity or 1,
the G, fifth above, is sounded by a 2/3 segment of the string sounding
C. The next fifth, G-D, takes us beyond the octave, and we find that
the D will be sounded by 4/9 (2/3 of 2/3 equals 4/9) of the entire
string, which fraction is less than half; so to keep within the bounds
of the octave, we must double this segment and make it sound the tone
D an octave lower, thus: 4/9 times 2 equals 8/9, the segment sounding
the D within the octave.
We may shorten the operation as follows: Instead of multiplying 2/3 by
2/3, giving us 4/9, and then multiplying this answer by 2, let us
double the fraction, 2/3, which equals 4/3, and use it as a multiplier
when it becomes necessary to double the segment to keep within the
octave.
We may proceed now with the twelve steps as follows:--
Steps--
1. 1C to 1G segment 2/3 for 1G
2. 1G " 1D Multiply 2/3 by 4/3, gives segment 8/9 " 1D
3. 1D " 1A " 8/9 " 2/3 " " 16/27 " 1A
4. 1A " 1E " 16/27 " 4/3 " " 64/81 " 1E
5. 1E " 1B " 64/81 " 2/3 " " 128/243 " 1B
6. 1B " 1F♯ " 128/243 " 4/3 " " 512/729 " 1F♯
7. 1F♯ " 1C♯ " 512/729 " 4/3 " " 2048/2187 " 1C♯
8. 1C♯ " 1G♯ " 2048/2187 " 2/3 " " 4096/6561 " 1G♯
9. 1G♯ " 1D♯ " 4096/6561 " 4/3 " " 16384/19683 " 1D♯
10. 1D♯ " 1A♯ " 16384/19683 " 2/3 " " 32768/59049 " 1A♯
11. 1A♯ " 1F " 32768/59049 " 4/3 " " 131072/177147 " 1F
12. 1F " 2C " 131072/177147 " 2/3 " " 262144/531441 " 2C
Now, this last fraction should be equivalent to 1/2, when reduced to
its lowest terms, if it is destined to produce a true octave; but,
using this numerator, 262144, a half would be expressed by
262144/524288, the segment producing the true octave; so the fraction
262144/531441, which represents the segment for 2C, obtained by the
circle of fifths, being evidently less than 1/2, this segment will
yield a tone somewhat sharper than the true octave. The two
denominators are taken in this case to show the ratio of the variance;
so the octave obtained from the circle of fifths is sharper than the
true octave in the ratio expressed by 531441 to 524288, which ratio is
called the _ditonic comma_. This comma is equal to one-fifth of a
half-step.
We are to conclude, then, that if octaves are to remain perfect, and
we desire to establish an equal temperament, the above-named
difference is best disposed of by dividing it into twelve equal parts
and depressing each of the fifths one-twelfth part of the ditonic
comma; thereby dispersing the dissonance so that it will allow perfect
octaves, and yet, but slightly impair the consonance of the fifths.
We believe the foregoing propositions will demonstrate the facts
stated therein, to the student's satisfaction, and that he should now
have a pretty thorough knowledge of the mathematics of the
temperament. That the equal temperament is the only practical
temperament, is confidently affirmed by Mr. W.S.B. Woolhouse, an
eminent authority on musical mathematics, who says:--
"It is very misleading to suppose that the necessity of temperament
applies only to instruments which have fixed tones. Singers and
performers on perfect instruments must all temper their intervals, or
they could not keep in tune with each other, or even with themselves;
and on arriving at the same notes by different routes, would be
continually finding a want of agreement. The scale of equal
temperament obviates all such inconveniences, and continues to be
universally accepted with unqualified satisfaction by the most eminent
vocalists; and equally so by the most renowned and accomplished
performers on stringed instruments, although these instruments are
capable of an indefinite variety of intonation. The high development
of modern instrumental music would not have been possible, and could
not have been acquired, without the manifold advantages of the
tempered intonation by equal semitones, and it has, in consequence,
long become the established basis of tuning."
NUMERICAL COMPARISON OF THE DIATONIC SCALE WITH THE TEMPERED
SCALE.
The following table, comparing vibration numbers of the diatonic scale
with those of the tempered, shows the difference in the two scales,
existing between the thirds, fifths and other intervals.
Notice that the difference is but slight in the lowest octave used
which is shown on the left; but taking the scale four octaves higher,
shown on the right, the difference becomes more striking.
|DIATONIC.|TEMPERED.| |DIATONIC.|TEMPERED.|
C|32. |32. |C|512. |512. |
D|36. |35.92 |D|576. |574.70 |
E|40. |40.32 |E|640. |645.08 |
F|42.66 |42.71 |F|682.66 |683.44 |
G|48. |47.95 |G|768. |767.13 |
A|53.33 |53.82 |A|853.33 |861.08 |
B|60. |60.41 |B|960. |966.53 |
C|64. |64. |C|1024. |1024. |
Following this paragraph we give a reference table in which the
numbers are given for four consecutive octaves, calculated for the
system of equal temperament. Each column represents an octave. The
first two columns cover the tones of the two octaves used in setting
the temperament by our system.
TABLE OF VIBRATIONS PER SECOND.
C |128. |256. |512. |1024. |
C♯ |135.61 |271.22 |542.44 |1084.89 |
D |143.68 |287.35 |574.70 |1149.40 |
D♯ |152.22 |304.44 |608.87 |1217.75 |
E |161.27 |322.54 |645.08 |1290.16 |
F |170.86 |341.72 |683.44 |1366.87 |
F♯ |181.02 |362.04 |724.08 |1448.15 |
G |191.78 |383.57 |767.13 |1534.27 |
G♯ |203.19 |406.37 |812.75 |1625.50 |
A |215.27 |430.54 |861.08 |1722.16 |
A♯ |228.07 |456.14 |912.28 |1824.56 |
B |241.63 |483.26 |966.53 |1933.06 |
C |256. |512. |1024. |2048. |
Much interesting and valuable exercise may be derived from the
investigation of this table by figuring out what certain intervals
would be if exact, and then comparing them with the figures shown in
this tempered scale. To do this, select two notes and ascertain what
interval the higher forms to the lower; then, by the fraction in the
table below corresponding to that interval, multiply the vibration
number of the lower note.
EXAMPLE.--Say we select the first C, 128, and the G in the same
column. We know this to be an interval of a perfect fifth. Referring
to the table below, we find that the vibration of the fifth is 3/2 of,
or 3/2 times, that of its fundamental; so we simply multiply this
fraction by the vibration number of C, which is 128, and this gives
192 as the exact fifth. Now, on referring to the above table of equal
temperament, we find this G quoted a little less (flatter), viz.,
191.78. To find a fourth from any note, multiply its number by 4/3, a
major third, by 5/4, and so on as per table below.
TABLE SHOWING RELATIVE VIBRATION OF INTERVALS BY IMPROPER FRACTIONS.
The relation of the Octave to a Fundamental is expressed by 2/1
" " " Fifth to a " " 3/2
" " " Fourth to a " " 4/3
" " " Major Third to a " " 5/4
" " " Minor Third to a " " 6/5
" " " Major Second to a " " 9/8
" " " Major Sixth to a " " 5/3
" " " Minor Sixth to a " " 8/5
" " " Major Seventh to a " " 15/8
" " " Minor Second to a " " 16/15
QUESTIONS ON LESSON XIII.
1. State what principle is demonstrated in Proposition II.
2. State what principle is demonstrated in Proposition III.
3. What would be the vibration per second of an exact (not
tempered) fifth, from C-512?
4. Give the figures and the process used in finding the vibration
number of the _exact_ major third to C-256.
5. If we should tune the whole circle of twelve fifths exactly as
detailed in Proposition III, how much too sharp would the last C
be to the first C tuned?
LESSON XIV.
~MISCELLANEOUS TOPICS PERTAINING TO THE PRACTICAL WORK OF TUNING.~
~Beats.~--The phenomenon known as "beats" has been but briefly alluded
to in previous lessons, and not analytically discussed as it should
be, being so important a feature as it is, in the practical operations
of tuning. The average tuner hears and considers the beats with a
vague and indefinite comprehension, guessing at causes and effects,
and arriving at uncertain results. Having now become familiar with
vibration numbers and ratios, the student may, at this juncture, more
readily understand the phenomenon, the more scientific discussion of
which it has been thought prudent to withhold until now.
In speaking of the unison in Lesson VIII, we stated that "the cause of
the waves in a defective unison is the alternate recurring of the
periods when the condensations and the rarefactions correspond in the
two strings, and then antagonize." This concise definition is
complete; but it may not as yet have been fully apprehended. The
unison being the simplest interval, we shall use it for consideration
before taking the more complex intervals into account.
Let us consider the nature of a single musical tone: that it consists
of a chain of sound-waves; that each sound-wave consists of a
condensation and a rarefaction, which are directly opposed to each
other; and that sound-waves travel through air at a specific rate per
second. Let us also remark, here, that in the foregoing lessons, where
reference is made to vibrations, the term signifies sound-waves. In
other words, the terms, "vibration" and "sound-wave," are synonymous.
If two strings, tuned to give forth the same number of vibrations per
second, are struck at the same time, the tone produced will appear to
come from a single source; one sweet, continuous, smooth, musical
tone. The reason is this: The condensations sent forth from each of
the two strings occur exactly together; the rarefactions, which, of
course, alternate with the condensations, are also simultaneous. It
necessarily follows, therefore, that the condensations from each of
the two strings travel with the same velocity. Now, while this
condition prevails, it is evident that the two strings assist each
other, making the condensations more condensed, and, consequently, the
rarefactions more rarefied, the result of which is, the two allied
forces combine to strengthen the tone.
In opposition to the above, if two strings, tuned to produce the same
tone, could be so struck that the condensation of one would occur at
the same instant with the rarefaction of the other, it is readily seen
that the two forces would oppose, or counteract each other, which, if
equal, would result in absolute silence.[G]
[G] When the bushing of the center-pin of the hammer butt becomes
badly worn or the hammer-flange becomes loose, or the condition of
the hammer or flange becomes so impaired that the hammer has too
much play, it may so strike the strings as to tend to produce the
phenomenon described in the above paragraph. When in such a
condition, one side of the hammer may strike in advance of the other
just enough to throw the vibrations in opposition. Once you may get
a strong tone, and again you strike with the same force and hear but
a faint, almost inaudible sound. For this reason, as well as that of
preventing excessive wear, the hammer joint should be kept firm and
rigid.
If one of the strings vibrates 100 times in a second, and the other
101, there will be a portion of time during each second when the
vibrations will coincide, and likewise a portion of time when they
will antagonize each other. The periods of coincidence and of
antagonism pass by progressive transition from one to the other, and
the portion of time when exactitude is attained is infinitesimal; so
there will be two opposite effects noticed in every second of time:
the one, a progressive augmentation of strength and volume, the other,
a gradual diminution of the same; the former occurring when the
vibrations are coming into coincidence, the latter, when they are
approaching the point of antagonism. Therefore, when we speak of one
beat per second, we mean that there will be one period of augmentation
and one period of diminution in one second. Young tuners sometimes get
confused and accept one beat as being two, taking the period of
augmentation for one beat and likewise the period of diminution. This
is most likely to occur in the lower fifths of the temperament where
the beats are very slow.
Two strings struck at the same time, one tuned an octave higher than
the other, will vibrate in the ratio of 2 to 1. If these two strings
vary from this ratio to the amount of _one_ vibration, they will
produce _two_ beats. Two strings sounding an interval of the fifth
vibrate in the ratio of 3 to 2. If they vary from this ratio to the
amount of _one_ vibration, there will occur _three_ beats per second.
In the case of the major third, there will occur _four_ beats per
second to a variation of _one_ vibration from the true ratio of 5 to
4. You should bear this in mind in considering the proper number of
beats for an interval, the vibration number being known.
It will be seen, from the above facts in connection with the study of
the table of vibration numbers in Lesson XIII, that all fifths do not
beat alike. The lower the vibration number, the slower the beats. If,
at a certain point, a fifth beats once per second, the fifth taken an
octave higher will beat twice; and the intervening fifths will beat
from a little more than once, up to nearly twice per second, as they
approach the higher fifth. Vibrations per second double with each
octave, and so do beats.
By referring to the table in Lesson XIII, above referred to, the exact
beating of any fifth may be ascertained as follows:--
Ascertain what the vibration number of the _exact_ fifth would be,
according to the instructions given beneath the table; find the
difference between this and the _tempered_ fifth given in the table.
Multiply this difference by 3, and the result will be the number of
beats or fraction thereof, of the tempered fifth. The reason we
multiply by 3 is because, as above stated, a variation of one
vibration per second in the fifth causes three beats per second.
_Example._--Take the first fifth in the table, C-128 to G-191.78, and
by the proper calculation (see example, page 147, Lesson XIII) we find
the exact fifth to this C would be 192. The difference, then, found by
subtracting the smaller from the greater, is .22 (22/100). Multiply
.22 by 3 and the result is .66, or about two-thirds of a beat per
second.
By these calculations we learn that the fifth, C-256 to G-383.57,
should have 1.29 beats: nearly one and a third per second, and that
the highest fifth of the temperament, F-341.72 to C-512, should be
1.74, or nearly one and three-quarters. By remembering these figures,
and endeavoring to temper as nearly according to them as possible, the
tuner will find that his temperament will come up most beautifully.
This is one of the features that is overlooked or entirely unknown to
many fairly good tuners; their aim being to get all fifths the same.
~Finishing up the Temperament.~--If your last trial, F-C, does not
prove a correct fifth, you must consider how best to rectify. The
following are the causes which result in improper temperament:
1. Fifths too flat.
2. Fifths not flat enough.
3. Some fifths correctly tempered and others not.
4. Some fifths sharper instead of flatter than perfect;
a condition that must be watched with vigilance.
5. Some or all of the strings tuned fall from the pitch
at which they were left.
From a little reflection upon these causes, it is seen that the last
trial may prove a correct fifth and yet the temperament be imperfect.
If this is the case, it will be necessary to go all over the
temperament again. Generally, however, after you have had a little
experience, you will find the trouble in one of the first two causes
above, unless it be a piano wherein, the strings fall as in Cause 5.
This latter cause can be ascertained in cases only where you have
started from a tuning pipe or fork. Sometimes you may find that the
temperament may be corrected by the alteration of but two or three
tones; so it is always well to stop and examine carefully before
attempting the correction. A haphazard attempt might cause much extra
work.
In temperament setting by our system, if the fifths are properly
tempered and the octaves are left perfect, the other intervals will
need no attention, and will be found beautifully correct when used in
testing.
The mistuned or tempered intervals are as follows:--
INTERVALS FLATTENED. INTERVALS SHARPENED.
The Fifth, slightly. The Fourth, slightly.
The Minor Third, The Major Third,
considerably. greatly.
The Minor Sixth, The Major Sixth,
considerably. greatly.
~Tuning the Treble.~--In tuning the treble, which is always tuned by
exact octaves, from their corresponding tones within the temperament,
the ear will often accept an octave as true before its pitch has been
sufficiently raised. Especially is this true in the upper octaves.
After tuning a string in the treble by its octave in the temperament,
test it as a fifth. For instance, after tuning your first string
beyond the temperament, 3C♯, test it as a fifth to 2F♯. If you are yet
uncertain, try it as a major third in the chord of A. The beats will
serve you as a guide in testing by fifths, up to about an octave and a
half above the highest tone of the temperament; but beyond this point
they become so rapid as to be only discernible as degrees of
roughness. The beats will serve as a guide in tuning _octaves_ higher
in the treble than the point at which the beats of the _fifth_ become
unavailable; and in tuning _unisons_, the beats are discernible almost
to the last tone.
The best method to follow in tuning the treble may be summed up as
follows: Tune the first octave with the beats as guides both in the
octave and in testing it by the fifth. If yet uncertain, test by
chords. Above this octave, rely somewhat upon the beats in the octave,
still use the fifth for testing, but listen for the pitch in the
extreme upper tones and not so much for the beats except in bringing
up unisons, in which the beats are more prominent.
In the extreme upper tones, the musical ear of the tuner is tried to
the utmost. Here, his judgment of correct harmonic relation is the
principal or only guide, while in the middle octaves the beats serve
him so faithfully, his musical qualifications being brought into
requisition only as a rough guide in determining pitch of the various
intervals. To tune by the beats requires a sharp ear and mental
discernment; to tune by pitch requires a fine musical ear and
knowledge of the simpler laws of harmony.
As stated above, the tuner will fail in many cases to tune his high
octaves sharp enough. Rarely, if ever, will a tuner with a good ear
leave the upper tones too sharp. Now, there is one more fact which is
of the utmost importance in tuning the treble: it is the fact that the
extreme upper octave and a half must be tuned slightly sharper than
perfect; if the octaves are tuned perfect, the upper tones of the
instrument will sound flat when used in scale and arpeggio passages
covering a large portion of the key-board. Begin to sharpen your
octaves slightly from about the seventeenth key from the last;
counting both black and white. In other words, begin to sharpen from
the last A♭ but one, in the standard scale of seven and a third
octaves of which the last key is C. Sharpen but slightly, and increase
the degree of sharpening but little as you proceed.
~Tuning the Bass.~--In tuning the bass, listen for the beats only, in
bringing up the octaves. It is sometimes well to try the string tuned,
with its fifth, but the octave in the bass should suffice, as the
vibrations are so much slower here that if you listen acutely the
octave beats will guide you.
It is not necessary to pull the strings higher than the pitch at which
they are to stand. Learn to pull them up gradually and in a way that
will "render" the string over the bridges, which is an easy thing to
do, the strings being so much heavier here than elsewhere. Never leave
a bass string the slightest amount too sharp. As flatness is so
obnoxious in the treble, just so is sharpness in the bass, so if there
must be any variation in any bass tone let it be flat; but aim at
perfect octaves throughout the bass.