THE EQUAL TEMPERAMENT.
As was suggested in the last chapter, it becomes necessary to effect
a compromise between the demands of true musical intonation and the
limitations of musical instruments, in order that the performance of
music may be made practicable. The equal temperament, now universally
employed, has only risen to its present commanding position within the
last century. It seems to have been first used by Johann Sebastian
Bach. Händel did not know it, and it struggled throughout the whole of
the eighteenth century with the mean-tone system.
Temperament systems were, however, invented and used long before this
period. Pythagoras, the Greek pre-Christian philosopher, was one of the
earliest experimenters along these lines. The method that he devised
has come down to us, and we are thus able to see wherein lies the
difference between it and the modern diatonic scale. Without going into
too much detail, we may note that the Pythagorean system recognizes
only two intervals; namely, the tone and semitone. The diatonic scale,
as we know, has a major tone, a minor tone and a diatonic semitone. The
Pythagorean scale contemplates perfect fifths and sharped thirds, and
is incapable of the effects of modern harmony.
The next attempt to adapt the necessary compromise in the interests
of practical music was introduced after the modern diatonic scale
had become the standard method of octave-division; that is to say,
some time in the fifteenth century. It has been variously called the
“mean-tone,” “mesotonic” and “vulgar” temperament. In this method
the tone is a mean or average between the major and minor tones of
the diatonic scale. The fifths are all flattened, while the thirds
are justly tuned. Such a system possesses both advantages and
disadvantages. On the one hand, the nearer and more frequently used
scales are purer and more agreeable; on the other hand, the remoter
scales are exceedingly dissonant; so much so, in fact, that they cannot
be employed with pleasure to either the performer or the hearer. So
long, however, as the music is written in the commoner scales the
mean-tone temperament, possessing the great advantage over other
methods of having pure thirds, is far more agreeable to the ear. In
fact, up till a few years ago it was not uncommon to find organs in
village churches in Europe that were still tuned according to this
system. The mean-tone system first made harmony, as we understand it,
practicable, but as the knowledge and imagination of composers widened,
the desire naturally arose to take advantage of the greater powers for
harmony that could alone be afforded by the unrestricted possession
of all possible scales. A substitute for the mean-tone system had,
therefore, to be found, and thus arose the modern and accepted method,
universally known as the Equal Temperament. By this method, which is at
the present time universal, the octave is divided into thirteen equally
distant semitones or half-steps. All distinctions between major and
minor tones and diatonic and chromatic semitones are swept away, and
it is assumed that the sound between any two sounds in the scale is
equally sharp and flat respectively to the sound immediately preceding
and following it.
This method, of course, implies a rearrangement of the whole scale,
for it is necessary to alter the precise pitch of every sound within
the compass of the octave in order that the equalization may be
effected. Thus it comes about that the equally tempered scale has only
one interval tuned purely. This interval naturally is the octave. All
the others require to be sharped or flatted in varying degrees. Every
chord, every interval, with one exception, therefore, is more or less
out of tune. The effect of this system of tempering cannot very well
be noted accurately upon the pianoforte, owing to the evanescence of
that instrument’s tone; but the organ often shows the dissonance of
certain intervals and chords in a most distressing manner. Perhaps
the worst of the defects of the Equal Temperament are exhibited in
the inability clearly to distinguish between true consonances and
true dissonances. Where the actual distinctions between the true
intervals are fused together it is impossible that there should be such
distinctions between them as the true scale shows, and, consequently,
we often are obliged to miss many delicate shades of comparative
consonance or dissonance that would be clearly exhibited in a scale in
which the intervals were represented with fidelity. We already know,
however, that no such method is at present possible, and we must fain
resign ourselves to the compromise that we have, and hope for better
things in the future. But at the same time, the Equal Temperament
possesses not a few positive virtues. As explained above, there can be
no difference between the sharp of a given tempered sound and the flat
of the tempered sound one whole step above the former. In other words,
the sharp of C in the Equal Temperament must be the same as the flat
of D, for these two sounds are assumed to be equally distant from the
sound which is between them, and the three are simply part of a series
of equal semitones. This being the case, the ambiguity that arises from
the identity of these sounds is very often found to be invaluable for
the purposes of quick and convenient modulation. There are instances
in which the connecting link between two modulations would entirely
be lost without the peculiar intonation that is afforded by equally
tempered sounds. It seems, in short, that the equal temperament,
imperfect and artificial as it is, cannot easily be replaced in the
existing states of our acoustical knowledge and of the mechanical
musical industries.
In order that the reader may more clearly realize the actual effects
of the Equal Temperament upon musical intonation, the following table
has been prepared, showing the differences of frequency between the
true sounds of the just chromatic scale and the corresponding tempered
sounds: (We are already familiar with the identity, in tempered
intonation, of the sharps and flats of adjacent degrees of the scale.)
C = 528 (Philharmonic Pitch).
True Scale. Equally Tempered Scale.
C 528 C 528
B 495 B 498-7/32
B flat 475-1/5 B flat--A sharp 470-7/20
A sharp 458-1/3 A 440
A 440 A flat--G sharp 417-1/20
A flat 422-2/5 G 391-11/20
G sharp 412-1/2 G flat--F sharp 373-7/20
G 396 F 342-4/10
G flat 380-4/25 E 332-17/20
F sharp 366-2/3 E flat--D sharp 313-19/20
F 352 D 296-7/20
E 330 D flat--C sharp 279-14/20
E flat 316-4/5 C 264
D sharp 309-11/24
D 297
D flat 285-3/25
C sharp 275
C 264
It would be without the province of our immediate purpose to enter
into any special discussion of the possibility of manufacturing
pianofortes that shall give pure intonation, as distinguished from
the tempered sounds that we have thus exhibited. We have already had
occasion to mention that the Equal Temperament has become so strongly
and intimately bound up with the performance of music, that the
majority of musicians are probably incapable of distinguishing between
the idea of pure as opposed to that of tempered musical sounds.
We have already pointed out, and reference to the various tables will
confirm the assertion, that the Equal Temperament imposes excessive
roughness of intonation upon very few of the musical intervals. Thus
the octave is pure, the fourths and fifths nearly so, and only the
seconds, thirds, sixths and sevenths are so rough as to be noticeable
to other ears than those of the professional pianoforte tuner.
Indeed it is very doubtful whether the musical public could ever be
universally educated to the point of appreciating the differences
between pure and equally-tempered fourths and fifths; while at the same
time it must be remembered that the second and seventh, at least, are
dissonances whether purely intoned or not.
We may properly question the actual advantage that the mechanical
attainment of just pianoforte intonation would produce; we may ask
ourselves what would be gained thereby for the cause of art, and the
answer does not appear to be other than that any conceivable benefit
must be so slight as to be practically negligible.