_A. From Major Tonic_
I Direct Relationships ii iii IV V vi
----------------------|------------------------------|-----|------|----|-----|--
| | | | | |
| Indirect through both | | | | |
| i and the second key | | iv v |
|------------------------------|-----|-----------------|--
| | | |
| | | |
Indirect, through i \ Indirect through the | | |
III[flat] VI[flat] \ second key | III VI
--|-----------|----------\---------------------------|--------------------------
| | \ |
Doubly indirect through the \ |
former indirect keys \ |
iii[flat] vi[flat] \ |
-------------------------------\---------------------|--------------------------
\ |
\ |
Artificial, direct II[flat] \ | VII & vii
-------------\----------------|--------------------------
\ |
Artificial, indirect[8] ii[flat] \ |
-----------------\------------|--------------------------
\ |
\ |
\ |
\ |
\ |
-------------------------------------------------\---|--------------------------
\ |
\ | IV[sharp] & iv[sharp] =
Unrelated \| V[flat] & v[flat]
-----------------------------------------------------\--------------------------
|\
| \
| \
Contradictory II VII[flat] & vii[flat]
--------------------------------------------------------------------------------
_B. From Minor Tonic_[9]
i Direct Relationships III iv v VI VII
----------------------|------------------------------|-----|------|----|-----|--
| | | | | |
| Indirect through both | | | | |
| i and the second key | IV V | |
| -----------------------------|-----------------|-----|--
\ | | |
\ | | |
Indirect, through i \ Indirect through the | | |
iii[sharp] vi[sharp] \ second key iii vi |
--|-----------|------------\-------------------------------------------------|--
| | \ /
| | \ /
Doubly indirect \ /
III[sharp] VI[sharp] | /
------------------------------|----------------------------------------/--------
\ /
\ /
Artificial, direct \ II[flat] |
\ |
\-----------------------------|------------
Artificial, indirect[10] \ ii[flat] VII[sharp] | & vii[sharp]
\---------------------------|------------
\ |
\ |
\ |
\ |
\ |
------------------------------------------------\------------------|-------------
\ |
\ IV[sharp] & | = V[flat] &
Unrelated \ iv[sharp | v[flat]
----------------------------------------------------\--------------|--------------
\ |
\ |
\ |
Contradictory[11] ii II vii[flat]
----------------------------------------------------------------------------------
VI. _Temperament and Enharmonic Changes._--As the facts of artistic
harmony increased in complexity and range, the purely acoustic
principles which (as Helmholtz has shown) go so far to explain
16th-century aesthetics became more and more inadequate; and grave
practical obstacles to euphonious tuning began to assert themselves. The
scientific (or natural) ratios of the diatonic scale were not interfered
with by art so long as no discords were "fundamental"; but when discords
began to assume independence, one and the same note often became
assignable on scientific grounds to two slightly different positions in
pitch, or at all events to a position incompatible with even tolerable
effect in performance. Thus, the chord of the diminished 7th is said to
be intolerably harsh in "just intonation," that is to say, intonation
based upon the exact ratios of a normal minor scale. In practical
performance the diminished 7th contains three minor 3rds and two
imperfect 5ths (such as that which is present in the dominant 7th),
while the peculiarly dissonant interval from which the chord takes its
name is very nearly the same as a major 6th. Now it can only be said
that an intonation which makes nonsense of chords of which every
classical composer from the time of Corelli has made excellent sense, is
a very unjust intonation indeed; and to anybody who realizes the
universal relation between art and nature it is obvious that the chord
of the diminished 7th must owe its naturalness to its close
approximation to the natural ratios of the minor scale, while it owes
its artistic possibility to the extremely minute instinctive
modification by which its dissonance becomes tolerable. As a matter of
fact, although we have shown here and in the article Music how
artificial is the origin and nature of all but the very scantiest
materials of the musical language, there is no art in which the element
of practical compromise is so minute and so hard for any but trained
scientific observation to perceive. If a painter could have a scale of
light and shade as nearly approaching nature as the practical intonation
of music approaches the acoustic facts it really involves, a visit to a
picture gallery would be a severe strain on the strongest eyes, as
Ruskin constantly points out. Yet music is in this respect exactly on
the same footing as other arts. It constitutes no exception to the
universal law that artistic ideas must be realized, not in spite of, but
by means of practical necessities. However independent the treatment of
discords, they assert themselves in the long run as transient. They
resolve into permanent points of repose of which the basis is natural;
but the transient phenomena float through the harmonic world adapting
themselves, as best they can, to their environment, showing as much
dependence upon the stable scheme of "just intonation" as a crowd of
metaphors and abstractions in language shows a dependence upon the rules
of the syllogism. As much and no more, but that is no doubt a great
deal. Yet the attempt to determine the point in modern harmony where
just intonation should end and the tempered scale begin, is as vexatious
as the attempt to define in etymology the point at which the literal
meaning of a word gives places to a metaphorical meaning. And it is as
unsound scientifically as the conviction of the typical circle-squarer
that he is unravelling a mystery and measuring a quantity hitherto
unknown. Just intonation is a reality in so far as it emphasizes the
contrast between concord and discord; but when it forbids artistic
interaction between harmony and melody it is a chimera. It is sometimes
said that Bach, by the example of his forty-eight preludes and fugues in
all the major and minor keys, first fixed the modern scale. This is true
practically, but not aesthetically. By writing a series of movements in
every key of which the keynote was present in the normal organ and
harpsichord manuals of his and later times, he enforced the system by
which all facts of modern musical harmony are represented on keyed
instruments by dividing the octave into twelve equal semitones, instead
of tuning a few much-used keys as accurately as possible and sacrificing
the euphony of all the rest. This system of _equal temperament_, with
twelve equal semitones in the octave, obviously annihilates important
distinctions, and in the most used keys it sours the concords and blunts
the discords more than unequal temperament; but it is never harsh; and
where it does not express harmonic subtleties the ear instinctively
supplies the interpretation; as the observing faculty, indeed, always
does wherever the resources of art indicate more than they express.
Now it frequently happens that discords or artificial chords are not
merely obscure in their intonation, whether ideally or practically, but
as produced in practice they are capable of two sharply distinct
interpretations. And it is possible for music to take advantage of this
and to approach a chord in one significance and quit it with another.
Where this happens in just intonation (in so far as that represents a
real musical conception) such chords will, so to speak, quiver from one
meaning into the other. And even in the tempered scale the ear will
interpret the change of meaning as involving a minute difference of
intonation. The chord of the diminished 7th has in this way four
different meanings--
[Illustration: Ex. 11.]
and the chord of the augmented 6th, when accompanied by the fifth, may
become a dominant 7th or vice versa, as in the passage already cited in
the coda of the slow movement of Beethoven's _B[flat] Trio, Op. 97._
Such modulations are called _enharmonic_. We have seen that all the more
complex musical phenomena involve distinctions enharmonic in the sense
of intervals smaller than a semitone, as, for instance, whenever the
progression D E in the scale of C, which is a minor tone, is identified
with the progression of D E in the scale of D, which is a major tone
(differing from the former as 8/9 from 9/10). But the special musical
meaning of the word "enharmonic" is restricted to the difference between
such pairs of sharps with flats or naturals as can be represented on a
keyboard by the same note, this difference being the most impressive to
the ear in "just intonation" and to the imagination in the tempered
scale.
Not every progression of chords which is, so to speak, spelt
enharmonically is an enharmonic modulation in itself. Thus a modulation
from D flat to E major looks violently enharmonic on paper, as in the
first movement of Beethoven's _Sonata, Op. 110_. But E major with four
sharps is merely the most convenient way of expressing F flat, a key
which would need six flats and a double flat. The reality of an
enharmonic modulation can be easily tested by transporting the passage a
semitone. Thus, the passage just cited, put a semitone lower, becomes a
perfectly diatonic modulation from C to E flat. But no transposition of
the sixteen bars before the return of the main theme in the scherzo of
Beethoven's _Sonata in E[flat], Op. 31_, No. 3, will get rid of the fact
that the diminished 7th (G B[flat] D[flat] E[natural]), on the dominant
of F minor, must have changed into G B[flat] D[flat] F[flat] (although
Beethoven does not take the trouble to alter the spelling) before it
could resolve, as it does, upon the dominant of A[flat]. But though
there is thus a distinction between real and apparent enharmonic
modulations, it frequently happens that a series of modulations
perfectly diatonic in themselves returns to the original key by a
process which can only be called an enharmonic circle. Thus the whole
series of keys now in practical use can be arranged in what is called
the circle of fifths (C G D A E B F[sharp] [= G[flat]] D[flat] A[flat]
B[flat] F C, from which series we now see the meaning of what was said
in the discussion of key-relationships as to the ambiguity of the
relationships between keys a tritone fourth apart). Now no human memory
is capable of distinguishing the difference of pitch between the keys
of C and B[sharp] after a wide series of modulations. The difference
would be perceptible enough in immediate juxtaposition, but after some
interval of time the memory will certainly accept two keys so near in
pitch as identical, whether in "just intonation" or not. And hence the
enharmonic circle of fifths is a conception of musical harmony by which
infinity is at once rationalized and avoided, just as some modern
mathematicians are trying to rationalize the infinity of space by a
non-Euclidian space so curved in the fourth dimension as to return upon
itself. A similar enharmonic circle progressing in major 3rds is of
frequent occurrence and of very rich effect. For example, the keys of
the movements of Brahms's _C Minor Symphony_ are C minor, E major,
A[flat] major (= G[sharp]), and C (= B[sharp]). And the same circle
occurs in the opposite direction in the first movement of his _Third
Symphony_, where the first subject is in F, the transition passes
directly to D[flat] and thence by exactly the same step to A (=
B[flat][flat]). The exposition is repeated, which of course means that
in "just intonation" the first subject would begin in G[flat][flat] and
then pass through a transition in E[flat][flat][flat] to the second
subject in C[flat][flat][flat]. As the development contains another
spurious enharmonic modulation, and the recapitulation repeats in
another position the first spurious enharmonic modulation of the
exposition, it would follow that Brahms's movement began in F and ended
in C sextuple-flat! So much, then, for the application of bad
metaphysics and circle-squaring mathematics to the art of music. Neither
in mathematics nor in art is an approximation to be confused with an
imperfection. Brahms's movement begins and ends in F much more exactly
than any wooden diagonal fits a wooden square.
The following series of musical illustrations show the genesis of
typical harmonic resources of classical and modern music.
[Illustration: Ex. 12.--Three concords (ionic, first inversion of
subdominant, and dominant of A minor, a possible 16th-century cadence
in the Phrygian mode).]
[Illustration: Ex. 13.--The same chords varied by a suspension (*).]
[Illustration: Ex. 14.--Ditto, with the further addition of a double
suspension (*) and two passing notes ([dagger][dagger]).]
[Illustration: Ex. 15.--Ditto, with a chromatic alteration of the
second chord (*) and an "essential" discord (dominant 7th) at
([dagger]).]
[Illustration: Ex. 16.--Ditto, with chromatic passing notes (**) and
appoggiaturas ([dagger][dagger]).]
[Illustration: Ex. 17.--The last two chords of Ex. 16 attacked
unexpectedly, the first appoggiatura (*) prolonged till it seems to
make a strange foreign chord before it resolves on the short note at
[Dagger], while the second appoggiatura ([dagger]) is chromatic.]
[Illustration: Ex. 18.--The same enharmonically transformed so as to
become a variation of the "dominant ninth" of C minor. The G[sharp] at
* is really A[flat], and [Dagger] is no longer a note of resolution,
but a chromatic passing-note._
_Definitions._
(Intended to comprise the general conceptions set forth in the above
article.)
1. _Musical sounds_, or _notes_, are sensations produced by regular
periodical vibrations in the air, sufficiently rapid to coalesce in a
single continuous sensation, and not too rapid for the mechanism of
the human ear to respond.
2. The _pitch_ of a note is the sensation corresponding to the degree
of rapidity of its vibrations; being _low_ or _grave_ where these are
slow, and _high_ or _acute_ where they are rapid.