SEPTeMBeR
11
THE MUSIC TRADE REVIEW
22, 1923
HOW CLOSELY CAN WE COME TO THEORETICAL
PERFECTION OF ACCURACY IN TUNING THE PIANO?
Mathematical Analysis of Temperament Gives Figures Which Show the Clo'se Approach to the
Limits of the Theory Which Can Be Achieved-For all Intents and Purposes Theory and
Practice May Be Said to Have Been Brought to an Equality
The controversy between the the orisl s and
the practicians is as old as philosophy herself.
Plato in the exqui s ite dialetical sword-play of
hi s Dialogues has over and over again shown
the absurdity of th e opinion wh ich is based
upon personal experience only, or upon a
tradition of personal experien ces and behind
which is no reason ed syste m of we ighed and
analyzed co nclusions based upon the gath
(;fed experiences of many a nd vario us minds
whose operations have extended over gr eat
spaces of time. The medieval philosophy did
ind eed carry the notion of lo gica l process as
in itself sufficient for the dis covery of truth,
to end s which we re cap ab le of obvious distor
tion in the hand s of second-rate intellects; but
the followers of Bacon went just as far in the
opposite direction and just as falsely concl ud ed.
For if th e rid dle of the Sphinx is not to be
answered by any pure ly mental exco gita tion,
neith e r is it to be found by grub bin g amid st
test-tubes; as the total failure of physical sc ience
to go beyond phenomena and their inter-re la
tions abundantly ev iden ces. Theory, in a word,
is the nec essa ry subst ratum and foundation o f
all. practice, and this is just as true in the ap
plied as in the abstract mathematical scie nc es
and in the arts founded upon them.
Of a ll the I.iberal arts, indeed, tho se . ",.:hich
rest upon discoveries and inventions in the
science .of mu sical sounds, the most perfectly
lend them selves to practice based sou ndly and
firmly epon the ba sis of scientific theory. Mu sic
herself, Heavenly Maid, rests upon a seat of
cOlllpletelyworked out and malhematically as
well as experimentally justified acoustic theory,
and the highest flights of inspirat ion of com
poser or interpreter are alik e groun ded upon
this solid basis. It is always absurd to talk
abo ut any form of practical activity being pos
sib le without a grou nd of theory on which to
rest; as absurd as to talk abo ut a language
without an alphabet. It is never more absurd
than in the case of musi cal art and of the other
arEs and crafts thereto al li ed.
The Que's tion Put
Of all the arts subs idiar y to music none ap
proaches in importance the art of mllsical
accordance or tuni ng. In no other is t he prac
tic al ap plicat ion so difficult and so painfully ac
quir ed. In no other, conve rse ly, is the the
oretical foundation so certai n, so acc urat ely de
limite·a an d so comp letely demonstrable. It is
therefore both strange and inter esting to dis-
cover that in no other oc cupa tion of like im
portance is there less agreement as to the de
pendence of prac tice upon theory, or le ss under
s tand ing amon g practitioners of how far their
p racti ce ca n be improved by th e acquirement of
exact knowledge. On the other ha nd, no a rt
of which I have any knowledge presents a more
interesting problem for research than does ours
in the qu est ion which may thus be phrased: .
Ho·ze far may one expect to go in adapting aile's
car alld hand to tile nicety of ad/ustme1lt which
the 1J/.otilenwtical expression of the equal tempera
ment de·m ullds i'
Some yea rs ago I wrote an art icle for a
I.ondon journal on " Tuning as Exact Science"
and argu ed that it is pos sib le to reduce the
cl a im s of theory and the lim itati ons of prac
tice nearly to equality; nearly e nough, in fact,
to ju st ify the sta tem ent that the tuner ca n in
his practice actually attain to an exactn ess far
closer than he usual ly himself imagines possible.
For this sta tem ent I was taken to task by an
emin ent Br iti sh authority, who had ye ars ago
made much apparatus fo r the great experi
m e nter Ellis, who translated Helmholtz a nd was
one of the most eminent of nineteenth centu ry
workers in sound. My cri tic believed that prac
tical tuning gives no more th an a very rough
app rox.i mati on to exactness and that the prac
tical tun er is in the vas t majority of cases quite
un ab le to do any better. To which I replied
that in at least on e set of expe rim ents made
some fifty years ago with the then bes t tuner
of lvlessrs. Broadwood in London the closeness
of hi s approach to th e acc ura cy of a mathe
matical temperament wa s ~imp ly amazing, and I
quoted Gr iesbach's figures in suppo rt of the
statement. Moreover, I said that if we take the
figures of beat-rates in the temperament octave,
from an y giv en pitch, and compare them with
the actua l res ults obtained by a ny first-class
tuner on a first-class piano, we find that the·
difference is too slight in most cases to be per
cept ibl e upon the piano, needing in str uments
fa r mor e dclica te to discl ose the discrepancy.
It is probable that in giving the title "Tuning
as E xac t Science" to my remarks I was leadin g
myself to an unintended deception; but my sole
intention was in this case to arg ue th at the
math ematical analysis of the temperament g ives
figures which can actually and pract icall y be
use d, and w hi c h, in fact, eve ry tuner should
use, becallse he really can come much closer
to them if he trie s than he usually admits.
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How close can on e come, then? The ques
tion is ve ry interestin g and every tun er ought
to think about it now and then. The more he
does think about it th e more anxiou s he wi ll
be to do the ver y finest work, because he will
the more be reali z in g ho\" this finest work of
his approaches to or recedes from the exa ctn ess
of the figures. Th at the fi g ures show what
aug·ht to be at ta ined m ea ns much; for the better
o ne knows what perfection r eally impli es, the
more likely one will be to reach p e rfection, at
las t.
Limitations of Pianoforte
In und ertak in g to answer the question, of
course, we must be gin with co nsidering the con
ditions within which we must work. In the
first place we must realize that the pianoforte is
by no means a perfect acoust ic in strument. It
Jl1ay be a very good, or 110t a very go od, piano.
The better it is the more nearly it wi ll enable
the tuner to do hi s best. In no case, however,
can the tuner get from a piano a durat ion of
sound, in anyo ne unison, of more than a few
seconds. It is proved experimentally that the
initial inten sity of a piano's sounds drops by
four-fifths of its st rength in the first tenth of a
second, and that it is safe to assign a duration,
for practical purpo ses, to p iano so und s of not
more than five seconds, even to the practiced
car of the tuner. Within th ese five seconds he
must ma'ke a ll his di sc riminator y estimates and
count his be at-rates. If he cannot do thi s with
a n approach t o exac tne ss his tuning cannot pos
sib ly be what I sho uld ca l1 "scientific. ·'
In fact, however, he can do thi s or something
ext remel y close to it. For instance, if we con
sider the octave F2 to F3, w hich is usually em
ployed for adjusting the temperament, we find
that at the int e rnati onal pitch the beat-rates of
the fi fths a re as foll ow s:
Here are
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