Grove's dictionary of music and musicians

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76 RESULTANT TONES RESULTANT TONES
EESULTANT TONES (Fr. Som rimltans
;
Ger. Combinationatone) are produced when any
two loud and sustained musical sounds are heard
at the same time. There are two kinds of
resultant tones, the Differential and the Summational.
The Differential tone ' ' is so called
because its number of vibrations is equal to the
di fferenoe between those of thegeneratingsounds.
The Summational tone ' '.
so called because its
number of vibrations is equal to the sum of
those of the generating sounds. The following
diagram shows the pitches of the differential
tones of the principal consonant intervals when
in perfect tune.
Generators.
Differentials,
If the interval be wider than an octave, as in
the last two examples, the differential is intermediate
between the sounds which produce it.
These tones can be easily heard on the ordinary
harmonium, and also on the' Organ. They are
not so distinct on the piano, because the sounds
of this instrument are not sustained. By
practice, however, the resultant tones can be
distinguished on the piano also.
Dissonant as well as consonant intervals produce
resultant tones. Takingthe minor Seventh
in its three possible forms the differentials are
as follows :
The first form of minor Seventh is obtained by
tuning two Fifths upwards (C-G-D) and then a
major Third downwards(D | Bl>) ; its differential
tone is /A|>, an exact major Third below C.
The second form is got by two exact Fourths
upwards (C-F-B|>) : the differential is then \Ab,
which is flatter than the previous IA\) by the
interval 35:36. The third form is the so-called
Harmonic Seventh on G, whose differential is G,
an exact Fourth below C. The marks \, /, here
used to distinguish notes which are confused
in the ordinary notation, will be found fully
explained under Temperament. We may
briefly remark that the acute sign # refers to
notes in an ascending series of Fifths, the grave
sign I to those in a descending series of Fifths.
Hitherto we have spoken only of the differential
tones which are produced by the fundamentals
or prime partial tones of musical sounds.
[See Partial Tones.] But a differential may
also arise from the combination of any upper
partial of one sound with any partial of the
other sound ; or from the combination of a
differential with a partial, or with another
differential. Thus the major Third C-E may
have the following differential tones :
All these tones are heard simultaneously ;
but
for convenience the differentials of the first,
second, third, and fourth orders are written
in notes of different length. We see, then,
that the number of possible resultant tones is
very great ; but only those which arise from
the primes of musical sounds are sufficiently
strong to be of practical importance.
In enabling the ear to distinguish between
consonant and dissonant intervals, the differential
tones are only less important than the
upper partials. Thus if the chord G-E-0 be
accurately tuned as 3 : 5 : 8, the differential of
G-G coincides with E, and that of E-G with G.
But if the intervals be tempered the differentials
are thrown out of tune, and give rise to beats.
These beats are very loud and harsh on the
ordinary harmonium, tuned in equal temperament.
Again, in the close triad C-E-G the
differentials of C-E and of E-G coincide and
give no beats if the intervals be in perfect tune.
On a tempered instrument the result is very
different.
If we take C to have 264 vibrations,
the tempered E has about 332^, and the
tempered G about 396^ vibrations. The differential
of C-E is then 68j, and that of E-G 63.
These two tones beat 5J times each second, and
thus render the chord to some extent dissonant.
In the minor triad, even when in just intonation,
several of the resultant tones do not fit in
with the no^es of the chord, although they may
be too far apart to beat. In the major triad, on
the contrary, the resultant tones form octaves
with the notes of the chord. To this difference
Helmholtz attributes the less perfect consonance
of the minor triad, and its obscured though not
inharmonious effect.
The origin of the differential tones has been
the subject of much discussion. Thomas Young
held that when beats became too rapid to be
distinguished by the ear, they passed into the
resultant tone. This view prevailed until the
publication in 1856 of Helmholtz's investigations,
in which many objections to Young's
theory were brought forward. To explain
what these objections are, it would be necessary