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39
The Music Trade Review
APRIL 17, 1926
The Technical Department—(Continued from page 38)
mental 162.94). On the other hand, we find the 43/100 beats per second, accounting for one beat
rate of the fourth partial of G sharp 2 to be in every (say) 2.36 seconds, or rather more than
821.17 (four times that of the fundamental G 2). two beats in five seconds. In the highest fifth
G 3, the beat
The difference between 821.17 and 814.70 is 6.40, of the temperament octave, G 3
which represents the 'six beats' heard when the rate is a little more than three beats per second.
"The fourths beat somewhat faster in the
interval is properly tempered.
nature of the case. The lowest fourth, C 2
Remaining Major Third
"The remaining major third within the octave F 2, beats at intervals of about 1.6 seconds, or
is Ab 2—C 3, which generates 8.14 beats per ten beats in sixteen seconds, equal to rather
second. This is shown by the fourth partial of more than three beats in five seconds. The
C 3, beats at intervals of
C 3, which has a frequency of four times the highest fourth, G 2
frequency of the fundamental C 2 or 4X258.65, about 1.1 seconds, or ten beats in eleven sec-
which equals 1034.6, and the fifth partial of A, onds, equal to a little less than one beat per
which has a frequency of five times its funda- second.
A Tuning Method
mental, or 5X205.29, which equals 1026.46. The
"In practice I should not think of tuning a
difference 1034.6—1026.46 equals 8.14 beats per
second, roughly equivalent to eight audible whole circle of fifths and fourths, but, rather,
have always followed the lesser hazard of tuning
beats, and accounting for them.
"Now, these three concrete examples should a half scale of each in opposite directions. Thus:
E, E
B, B — F sharp, F sharp
enable anyone to work out all the remaining Fifths, A
major thirds within the octave and to master —C sharp, C sharp—G sharp, G sharp—D sharp,
the theory for all other consonant intervals. It and then by fourths, D sharp—A, A—D, D—G,
is necessary to bear in mind that in the minor G—C, C—F, F—Bb, Bb—Eb, making as the goal
thirds the coincident partial tones are the sixth the unison D sharp—Eb.
"Then I am ready to apply all the thirds and
of the lower and the fifth of the higher, member
sixths tests in the category, with reasonable
of the interval.
satisfaction where conditions are not too ad-
Minor Third
"We work out the minor thirds as follows: verse.
"This routine method fastened itself on my
Beginning with C 2, we have C 2
Eb 2,
whose rates in just intonation are to each other practice because it saves time, but sometimes
5:6. (C 2) 129.32X6 is just the same as (E 2) for novelty I have tuned from A to E and D
155.49X5, or 775.95. But when we take the below and then tuned major and minor thirds
tempered tones we find that tempered E b is above and below each of these, and the minor
153.79, and five times that is 768.97, which is third above A; then B a major third above G
6.98 less than the sixth partial of C 2. This and sixth above E. A sharp is then tuned as
accounts for the 'seven beats' required to be major third above F sharp and checked as
fourth above F.
heard as set forth by Mr. Hale.
Need of Understanding
"I shall give' one more illustration of a minor
"Equipped with an understanding of the
third beat rate. Take the minor third A2
C3.
The frequency of A is 217.5, and its sixth partial, nature of the beats, it is immaterial which way
therefore, stands at six times as much, or one goes, whether by thirds and sixths or by
1305.00. The frequency of tempered C 3 is fifths and fourths. One may tune principally
258.65, and that of its fifth partial is, therefore, by either and check up by the other. But one
1293.26, which gives us 11.74 difference between must know just what one is doing, and also
the two partial tone frequencies, and this ac- just how to do it. Then one is safe just so far
counts for the eleven or twelve audible beats.
"In the case of the major sixth interval the
coincident partial tones are the fifth of the lower
and the third of the upper member of the in-
terval. Taking the interval C 2
A 2, we have
C 2 with 129.65 and A 2 with 217—5 v. p. s.
129.65X5=646.6
217.5 X3=652.5
Difference= 5.9 v. p. s., ac-
counting for the 'six audible beats' required.
"In the case of the minor sixths the coincident
partial tones are the eighth of the lower and
the fifth of the higher. Take the case of C 2
Ab 2, of which the respective frequencies are
129.65:
129.32X8=1034.60
205.29X5=1026.45
Difference= 8.15 v. p. s., which
accounts for the 'eight beats' required.
Fifths and Fourths
"Let me now illustrate the working of the
principle in the case of the perfect fifths. Let
us take the interval C 2
G 2. Then the third
partial of C 2 will correspond with the second
partial of G 2, and the calculation will be
with tempered G
(C)
129.32X3=387.97
(G)
193.77X2=387.54
Difference=
as one has mastered the art through experience,
intelligence and fidelity.
"J. C. MILLER, Lincoln, Neb."
Comment
First of All, Please Note Two Things: Mr.
Miller bases his figures on international pitch
A 435. The substitution of the standard 1925
pitch A 440 for this other will not make any
difference sufficiently great to be perceptible to
the tuner's ear in that region of the piano where
the temperament is always laid down. There-
fore I have not changed the figures, since there
is no other difference, and they would be pro-
portionately the same at any pitch.
Another point. In order to reduce the com-
plexity of the calculations I have kept them
down to two places of decimals, although Mr.
Miller accurately worked them out farther.
Therefore some of the multiplications and sub-
tractions may seem to be inaccurate after the
first decimal point. This, however, means
nothing for practical purposes.
An Analysis
I might point out that in my book, "Modern
Piano Tuning," I have given a lengthy analysis
of the matter of coincident beats and their use
in tuning, usmg, in fact, as the basis of the
explanation the valuable set of tables long ago
prepared by Mr. Miller, at my suggestion, in
which the beat rates at international pitch are
given for every interval used in tuning, from
C 2— to C 4. When the tables were prepared
and the book written the international pitch
A 435 was official. But, as I have said, this
really makes very little difference—in fact, for
the present purpose no practical (i. e., audible)
difference at all. In my book I have also very
carefully explained just how and why beats arise
only between these coincident partials, and not
otherwise.
Correspondence
is solicited and should be addressed to William
Braid White, 5149 Agatite avenue, Chicago, 111.
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