Music Trade Review

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PIANO FACTORY and
PIANO SERVICING
DR. W M . BRAID WHITE
Technical Editor
SCIENCE OF TUNINC
and the
FUTURE OF THE ART
of music, owing to the fact that the key-
board instruments which have come to domi-
nate the musical art are limited to twelve
tones in each octave. From this I have gone
on to describe the system of equal tempera-
ment which during more than a century has
prevailed over all other compromise systems
of tuning. I have shown how this works
out mathematically, and have then drawn up
tables showing the arithmetical ratios actu-
ally set up, with the individual frequencies
DR. WM. BRAID WHITE
for the standard pitch. Directions for ap-
plying these tables have been appended.
UITE a long time has elapsed since
It seems that some readers still find diffi-
my last talk in these columns on
any matter connected with the culty in understanding the demonstration.
noble art of tuning. I am re- Mr. Fink has read on page 77, 2nd edition,
minded of this rather insistently by two let- how it is that although twelve successive
ters which lie on my desk awaiting reply. ascending fifths coincide on the keyboard
with seven successive ascending octaves be-
One of them is from Robert A. Fink of
Cleveland, who evidently has been studying ginning from the same tone, they are not
Modern Piano Tuning. The question he equal in sound, but exceed the octaves in
asks is of considerable interest, if only be- the ratio 531441 to 524288. Parallel consid-
cause it suggests the possibility that others erations are shown to obtain in the fourths,
may have misunderstood my argument in the twelve of which successively ascending from
first chapters of that book. Those who have a given tone are equal in width but not
equal in sound to five octaves ascending suc-
studied Modern Piano Tuning will remember
that in it I have attempted to set forth a very cessively from the same tone. The differ-
ence is this time in favor of the octaves,
definite rule for laying the equal tempera-
and the ratio is 192:144; or conversely it
ment. I have done this by first investigat-
ing the musical scale in its origin and trac- may be said that the fourth series falls short
ing out the natural pure relations among its of the corresponding octave series in the
various steps. I have then shown how it is ratio 144 : 192. Mr. Fink then has turned
impossible to maintain these pure relations to the tables showing the rates of beats be-
in the practical composition and performance tween the members of intervals correctly
tuned in equal temperament and finds dis-
crepancies between what these tables con-
tain and the calculations just mentioned. He
wants an explanation. This I gladly give.
Q
Estate/
FORCE OF HABIT
I suppose the fact to be that we have all
/HANUFACTORER
WHERE CAN YOU GET
PLAYER ACTION
REPAIRS and SUPPLIES
BUCKSKIN.
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Deep River, Conn.
1049—3rd St.
NORTH BERGEN, N. J.
Tel.: 7—4367
THE
become so thoroughly accustomed to taking
tempered intonation for granted as to forget
that the diatonic scale was originally con-
ceived, very naturally, in its obvious pure
relations. Everybody knows that if you have
a tuning fork giving 100 vibrations per sec-
ond (or cycles, as they are now more usually
called) and then another fork giving 150
vibrations per second, the two will sound ex-
actly a perfect fifth apart. Everybody knows
too that if these ratios, instead of being 100:
150, which is 2 : 3 or 1 : V/2, they were 100:
133.33, which is 3 :4 or 1 : 1J4 the two forks
would sound a perfect fourth apart. The
ratios between vibration speeds which would
give the other intervals of the scales are
equally familiar. What is overlooked so
often is that these pure intervals which are
so soothing and delightful to the ear are very
seldom realized in the actual practical art of
music, owing to the fact, which I have shown
in Modern Piano Tuning, that unless we
have more than twelve tones available in
each octave, we cannot maintain these pure
relations. It is to show that this is unfor-
tunately true that I have spoken of the dis-
crepancy in pitch between twelve successive
fifths and seven successive octaves taken from
the same tone, and between twelve fourths
and five octaves taken in the same way.
Simple arithmetic will demonstrate the cor-
rectness of my figures. Suppose we start at
the lowest C on the piano, (C 4 ) calling the
pitch 32.5 (which is exact enough). Let
us then multiply this pitch by 3/2, so as to
obtain the first fifth above in pure relation.
•Then let us continue until w T e have twelve
of these successive fifths piled up on top of
each other. Actually the key on the key-
board which we shall reach will be Css.
The pitch of this in round numbers is 4186
MUSIC
TRADE
REVIEW,
February, 1932
21