Music Trade Review
Issue: 1913 Vol. 57 N. 21 - Page 12
Music Trade Review -- © mbsi.org, arcade-museum.com -- digitized with support from namm.org
12
THE
MUSIC TRADE REVIEW
OuTTECHNICAL DEPARTMENT
CONDUCTED BY WILLIAM BRAID WHITE.
INTERESTING COMMENTS ON THE MILLER TABLES
Which Appeared in This Department in Last Week's Review—Mr. Miller Has Given the Be-
ginning of a Technical Formula for Use in the Work of Tuners.
I feel really that the opportunity cannot be per-
mitted to pass to say a few words about those
Miller tables which appeared in last week's Re-
view. May I begin by remarking that Mr. Miller
has done a very large thing for all of us tuners.
He has given us the beginning of a technical
formula manual for use in our work. He has
begun the work of providing for the piano tunei
a text-book of formula comparable with the
monumental works available for the use of civil
and mechanical engineers. The fact of the matter
quite soberly is that hitherto we men of the piano
trade have not been well treated by those who are
recognized as real simon-pure engineers. Yet 1
am sure that we should not at all be exaggerating
the state of the case if we insisted that we, too, are
engineers; acoustic engineers. And it can hardly
be doubted that we need as many standard tables
of formula as we can get. Hence the very great
value of the work of such men as Miller. I don't
know how thoroughly the mass of my readers un-
derstands the value and the necessity of this sort
of work. But it is plain enough that piano tuning
' will r be a very inexact art until everybody does
recognize and appreciate this essentiality.
There is, however, a good deal more for me to
do than merely to lavish well deserved praise on
the broad shoulders of Brother (I had almost
said "Father") Miller. I feel that 'his work, as
published last week, requires at my hands the ad-
dition of a supplement. And that supplement
seems to me to be necessary for the purpose of
bringing to those readers, who do not understand
the value of the Miller tables, some appreciation
of their worth and of their practical use in the
practical work of piano tuning.
Equal temperament means that we must have a
standard of measurement; a sort of acoustic foot
rule that we can apply to our problems and thus
read them off. As we all know, the audible beats
that arise between two members of an interval
when concord is disturbed, furnish us with our
ear-rule, whereby we are enabled to know what
we are doing. But I find more and more often as
time goes on that the mass of tuners have but a
very vague notion of the reason for the existence
of beats, or of any system for accurately esti-
mating their frequency. Mr. Miller tells us, quite
correctly, that beats arise between disturbed
unisons. From that correct premise he proceeds
to show that the unisons in question are to be
found, in any interval, among the partial tones of
each member thereof. But it seems to me that a
little more explanation might be useful here.
Beats arise when two systems of waves, of
definite frequency ratio, are sounded together in
such a way that one system is slightly thrown out
of ratio with the other. If the ratio be as simple
as 1 : 1 ; that is, if both systems have the same
frequency, and if this then be slightly changed,
the development of beats may be easily observed.
The reason for this development lies in the fact that
when two systems are thus started going, with one,
say, one vibration per second slower than the other
the orderly progression with which they start will
soon be disturbed. If both are in the same phase
when they start, one will soon drop behind, and at
FAUST SCHOOL OF TUNING
Piano, Player-Piano, Pipe and Reed Organ Tuning and Re-
pairing, alto Regulating, Voicing, Varnishing and Polishing
This formerly was the tuning department of the New Eng-
land Conservatory of Music, and Oliver C. Faust was head
of that department for 20 years previous to its discontinu-
ance.
Courses in mathematical piano scale construction and
drafting of same have been added.
Pupils have daily practise in Chickering A Sons' factory.
Year Book lent free apon request
17-29 GAINSBOROUGH ST., BOSTON, MASS.
the conclusion of one second one will be a whole vi-
bration behind the other. In other words, there will
have been a change of phase, and again another
change, bringing the two systems back into the
same phase. But while this is going on the
change of phase has produced a condition where,
for a moment, one system is completely at its
crest while the other is completely in its trough,
followed by an immediate crossing over of the two
systems and a reversal of the phase. Now, it is
obvious that the result of this condition must be
the alternate production, once in each second, of
an augmentation of sound followed by a period of
silence. The latter corresponds to the exact op-
position of phase and the other to the piling of
crest on crest or trough below trough, as the case
may be. The two conditions must necessarily fol-
low one another immediately and the result must
be, one in each second, a moment of increased
sound followed by a moment of silence. The two
together make up the peculiar rising and falling of
sound that we call a "beat."
Now, since this is the explanation of the exist-
ence of a beat, it must be perfectly plain that such
manifestations can only be appreciable when the
wave systems involved are in such definite ratios
as may be expressed by 1 : 1, 1 : 2 or multiples
thereof. It is undoubted that, in any conceivable
ratio, beats will come to exist, but considering the
peculiarly complex conditions of their origin, it is
more than doubtful whether they could possibly
become audible amidst the fleeting vibrations of
piano strings, except through the medium of
unisonal wave systems. That is to say, beats are
hard enough to hear anyway, and it would prob-
ably be impossible to distinguish any that might
arise through ratios more complex than those of
the unison or the octave.
That being the case, we begin to see that we
cannot hear beats in intervals like the third,
fourth, fifth, sixth or seventh, unless we hear
them through some form of unison. Now it is
clear that such unisons are not to be found in the
fundamentals of such intervals, for these are not
unisonal. But it is possible to find such unisons in
the partial tones which each gives off through the
medium of the compound vibration characteristic
of piano strings. If we take any two tones that
we care to select and run through the list of their
partial tones we shall finally arrive at some one
common to each. Of course, while this is true of
any conceivable pair of tones, no matter how far
apart, it is practically true only within definite
limits. But for our purposes—that is, for tuning
—we can find coincidental or communal partial
tones very easily in any interval with which we
have to deal.
Now these coincidental partial tones will clearly
Piano Trade School
be absolutely at the same pitch when the interval
itself is in concord. Just as soon, however, as
we begin to sharp or flat one or other of the mem-
bers of the interval, the coincidental partials are
thrown out of unison and the corresponding beats
are heard. The number of the beats is, naturally,
equivalent to the number of phase reversals in each
second between the two wave systems representing
the two tones. Hence it is equal to the difference
in frequency between the two coincidental partials;
not between fundamentals.
Hence again, if we know how many beats
should arise between the coincidental partial tones
in a rightly tempered interval, we have only to
temper so as to produce that number of beats
and we shall be sure that the tempering is correct.
So when the Miller tables show us the right
number of beats for a given interval in equal
temperament, we have only to temper accordingly
to be sure we are correct for that particular pitch.
If the number of beats is less than one per
second, we can test our work by observing the
elapsed time between beats as shown in the tables.
Perhaps the method of calculating should be
spoken of briefly. It is plain that the thing to do
in each interval is first to find the coincidental
partial tones. This is readily done by the simple
method of comparison. Take, for example, the
imaginary tones C = 200 and G = 300. Extend-
ing the partials of C we find them to be 200, 400,
600, 800, 1,000 and so on. The partials of G are,
in the same way, 300, G00, 900, 1,200 and so on. By
comparison we at once find that the 3d partial of
C and the 2d partial of G are coincidental. Beats
then will arise when one of these is slightly thrown
out of unison. And similar comparisons may be
made for all the other intervals.
Now, to get the pitch of the fundamental tones
in an equal temperament diatonic-chromatic scale,
we simply multiply the starting tone by the 12th
root of 2, thus getting the next ascending equal
tempered semitone. But continuing this process
of multiplication by semitones ascending or by
dividing similarly for descending tones, we can find
the equal tempered pitch of as many as we please.
We then have but to take the various intervals,
find the coincidental partiajs, multiply these out,
take the difference between them, and that gives
us the number of beats that will be heard in one
second if the interval is rightly tempered. If that
number of beats be not heard, the interval is not
rightly tempered and that is all there is to it.
One last word: Is all this worth while? As-
suredly it is. You may not be able to temper
exactly according to the beat-rates set down in the
tables, but you can get pretty close to them. And
the big thing is that you have here an absolutely
exact formula, which tells you what the exactly
right thing is. Nothing is left to guess work. All
is rigidly accurate. That is why you need the
Miller tables.
Communications for this department should be
addressed to the Editor, Technical Department,
The Music Trade Review.
Established 1901
The Hodgin Piano Co.' Greensboro, N. C, has
been incorporated with capital stock of $25,000,
business to be commenced when $5,000 of the
stock has been subscribed. The incorporators are
John A. Hodgin, L. M. Foust and E. C. Hamilton.
Polk's School of Tuning
If you desire a man for any department of
our service, forward your advertisement to ua
and it will be inserted free of charge.
Piano, .Player-Piano and Organ tuning, repairing,
regulating, and voicing. Most thoroughly equipped
piano trade school in U. S. 600 graduates; private
instruction; diplomas awarded. Endorsed by lead-
ing piano manufacturers and dealers. School en-
tire year. Positions secured. Free illustrated cata-
logue. Address
Box 293
Valparaiso. Ind.
The Tuners' Magazine
A Monthly Journal, Devoted to the Joint Interests of
the Manufacturers and Tuners of
Musical Instruments.
SVMNER L. BALES, Editor and Proprietor
No. 1 Saa Rafael, Cincinnati, Ohio
lamed the Pint of the Month
Terata, $1.00 per Y«w.
NORTHWESTERN OHIO SCHOOL
OF
PIANO TUNING
Catalog
D. 0. BETZ, Dir., Ada, 0.
12
THE
MUSIC TRADE REVIEW
OuTTECHNICAL DEPARTMENT
CONDUCTED BY WILLIAM BRAID WHITE.
INTERESTING COMMENTS ON THE MILLER TABLES
Which Appeared in This Department in Last Week's Review—Mr. Miller Has Given the Be-
ginning of a Technical Formula for Use in the Work of Tuners.
I feel really that the opportunity cannot be per-
mitted to pass to say a few words about those
Miller tables which appeared in last week's Re-
view. May I begin by remarking that Mr. Miller
has done a very large thing for all of us tuners.
He has given us the beginning of a technical
formula manual for use in our work. He has
begun the work of providing for the piano tunei
a text-book of formula comparable with the
monumental works available for the use of civil
and mechanical engineers. The fact of the matter
quite soberly is that hitherto we men of the piano
trade have not been well treated by those who are
recognized as real simon-pure engineers. Yet 1
am sure that we should not at all be exaggerating
the state of the case if we insisted that we, too, are
engineers; acoustic engineers. And it can hardly
be doubted that we need as many standard tables
of formula as we can get. Hence the very great
value of the work of such men as Miller. I don't
know how thoroughly the mass of my readers un-
derstands the value and the necessity of this sort
of work. But it is plain enough that piano tuning
' will r be a very inexact art until everybody does
recognize and appreciate this essentiality.
There is, however, a good deal more for me to
do than merely to lavish well deserved praise on
the broad shoulders of Brother (I had almost
said "Father") Miller. I feel that 'his work, as
published last week, requires at my hands the ad-
dition of a supplement. And that supplement
seems to me to be necessary for the purpose of
bringing to those readers, who do not understand
the value of the Miller tables, some appreciation
of their worth and of their practical use in the
practical work of piano tuning.
Equal temperament means that we must have a
standard of measurement; a sort of acoustic foot
rule that we can apply to our problems and thus
read them off. As we all know, the audible beats
that arise between two members of an interval
when concord is disturbed, furnish us with our
ear-rule, whereby we are enabled to know what
we are doing. But I find more and more often as
time goes on that the mass of tuners have but a
very vague notion of the reason for the existence
of beats, or of any system for accurately esti-
mating their frequency. Mr. Miller tells us, quite
correctly, that beats arise between disturbed
unisons. From that correct premise he proceeds
to show that the unisons in question are to be
found, in any interval, among the partial tones of
each member thereof. But it seems to me that a
little more explanation might be useful here.
Beats arise when two systems of waves, of
definite frequency ratio, are sounded together in
such a way that one system is slightly thrown out
of ratio with the other. If the ratio be as simple
as 1 : 1 ; that is, if both systems have the same
frequency, and if this then be slightly changed,
the development of beats may be easily observed.
The reason for this development lies in the fact that
when two systems are thus started going, with one,
say, one vibration per second slower than the other
the orderly progression with which they start will
soon be disturbed. If both are in the same phase
when they start, one will soon drop behind, and at
FAUST SCHOOL OF TUNING
Piano, Player-Piano, Pipe and Reed Organ Tuning and Re-
pairing, alto Regulating, Voicing, Varnishing and Polishing
This formerly was the tuning department of the New Eng-
land Conservatory of Music, and Oliver C. Faust was head
of that department for 20 years previous to its discontinu-
ance.
Courses in mathematical piano scale construction and
drafting of same have been added.
Pupils have daily practise in Chickering A Sons' factory.
Year Book lent free apon request
17-29 GAINSBOROUGH ST., BOSTON, MASS.
the conclusion of one second one will be a whole vi-
bration behind the other. In other words, there will
have been a change of phase, and again another
change, bringing the two systems back into the
same phase. But while this is going on the
change of phase has produced a condition where,
for a moment, one system is completely at its
crest while the other is completely in its trough,
followed by an immediate crossing over of the two
systems and a reversal of the phase. Now, it is
obvious that the result of this condition must be
the alternate production, once in each second, of
an augmentation of sound followed by a period of
silence. The latter corresponds to the exact op-
position of phase and the other to the piling of
crest on crest or trough below trough, as the case
may be. The two conditions must necessarily fol-
low one another immediately and the result must
be, one in each second, a moment of increased
sound followed by a moment of silence. The two
together make up the peculiar rising and falling of
sound that we call a "beat."
Now, since this is the explanation of the exist-
ence of a beat, it must be perfectly plain that such
manifestations can only be appreciable when the
wave systems involved are in such definite ratios
as may be expressed by 1 : 1, 1 : 2 or multiples
thereof. It is undoubted that, in any conceivable
ratio, beats will come to exist, but considering the
peculiarly complex conditions of their origin, it is
more than doubtful whether they could possibly
become audible amidst the fleeting vibrations of
piano strings, except through the medium of
unisonal wave systems. That is to say, beats are
hard enough to hear anyway, and it would prob-
ably be impossible to distinguish any that might
arise through ratios more complex than those of
the unison or the octave.
That being the case, we begin to see that we
cannot hear beats in intervals like the third,
fourth, fifth, sixth or seventh, unless we hear
them through some form of unison. Now it is
clear that such unisons are not to be found in the
fundamentals of such intervals, for these are not
unisonal. But it is possible to find such unisons in
the partial tones which each gives off through the
medium of the compound vibration characteristic
of piano strings. If we take any two tones that
we care to select and run through the list of their
partial tones we shall finally arrive at some one
common to each. Of course, while this is true of
any conceivable pair of tones, no matter how far
apart, it is practically true only within definite
limits. But for our purposes—that is, for tuning
—we can find coincidental or communal partial
tones very easily in any interval with which we
have to deal.
Now these coincidental partial tones will clearly
Piano Trade School
be absolutely at the same pitch when the interval
itself is in concord. Just as soon, however, as
we begin to sharp or flat one or other of the mem-
bers of the interval, the coincidental partials are
thrown out of unison and the corresponding beats
are heard. The number of the beats is, naturally,
equivalent to the number of phase reversals in each
second between the two wave systems representing
the two tones. Hence it is equal to the difference
in frequency between the two coincidental partials;
not between fundamentals.
Hence again, if we know how many beats
should arise between the coincidental partial tones
in a rightly tempered interval, we have only to
temper so as to produce that number of beats
and we shall be sure that the tempering is correct.
So when the Miller tables show us the right
number of beats for a given interval in equal
temperament, we have only to temper accordingly
to be sure we are correct for that particular pitch.
If the number of beats is less than one per
second, we can test our work by observing the
elapsed time between beats as shown in the tables.
Perhaps the method of calculating should be
spoken of briefly. It is plain that the thing to do
in each interval is first to find the coincidental
partial tones. This is readily done by the simple
method of comparison. Take, for example, the
imaginary tones C = 200 and G = 300. Extend-
ing the partials of C we find them to be 200, 400,
600, 800, 1,000 and so on. The partials of G are,
in the same way, 300, G00, 900, 1,200 and so on. By
comparison we at once find that the 3d partial of
C and the 2d partial of G are coincidental. Beats
then will arise when one of these is slightly thrown
out of unison. And similar comparisons may be
made for all the other intervals.
Now, to get the pitch of the fundamental tones
in an equal temperament diatonic-chromatic scale,
we simply multiply the starting tone by the 12th
root of 2, thus getting the next ascending equal
tempered semitone. But continuing this process
of multiplication by semitones ascending or by
dividing similarly for descending tones, we can find
the equal tempered pitch of as many as we please.
We then have but to take the various intervals,
find the coincidental partiajs, multiply these out,
take the difference between them, and that gives
us the number of beats that will be heard in one
second if the interval is rightly tempered. If that
number of beats be not heard, the interval is not
rightly tempered and that is all there is to it.
One last word: Is all this worth while? As-
suredly it is. You may not be able to temper
exactly according to the beat-rates set down in the
tables, but you can get pretty close to them. And
the big thing is that you have here an absolutely
exact formula, which tells you what the exactly
right thing is. Nothing is left to guess work. All
is rigidly accurate. That is why you need the
Miller tables.
Communications for this department should be
addressed to the Editor, Technical Department,
The Music Trade Review.
Established 1901
The Hodgin Piano Co.' Greensboro, N. C, has
been incorporated with capital stock of $25,000,
business to be commenced when $5,000 of the
stock has been subscribed. The incorporators are
John A. Hodgin, L. M. Foust and E. C. Hamilton.
Polk's School of Tuning
If you desire a man for any department of
our service, forward your advertisement to ua
and it will be inserted free of charge.
Piano, .Player-Piano and Organ tuning, repairing,
regulating, and voicing. Most thoroughly equipped
piano trade school in U. S. 600 graduates; private
instruction; diplomas awarded. Endorsed by lead-
ing piano manufacturers and dealers. School en-
tire year. Positions secured. Free illustrated cata-
logue. Address
Box 293
Valparaiso. Ind.
The Tuners' Magazine
A Monthly Journal, Devoted to the Joint Interests of
the Manufacturers and Tuners of
Musical Instruments.
SVMNER L. BALES, Editor and Proprietor
No. 1 Saa Rafael, Cincinnati, Ohio
lamed the Pint of the Month
Terata, $1.00 per Y«w.
NORTHWESTERN OHIO SCHOOL
OF
PIANO TUNING
Catalog
D. 0. BETZ, Dir., Ada, 0.
Future scanning projects are planned by the International Arcade Museum Library (IAML).