Enharmonic
modulations with duble diminished chords are very often processed like
this:
-
A note is added to a diatonic major scale degree ann
in the key in which we are at first; this note we hear at first as minor
seventh; it seems that the major traid therefore is turned into a dominant
seventh chord
-
When the chord resolves it turns out that the (perceived) minor seventh
was no minor seventh after all, but an augmented sixth instead (or: it
turns out that the seventh has changed to an augmented sixth). Which
means that the chord turns out to be an Italian or German augmented ( #IVdv6
or #IVdv6/5 ) and to resolve to the dominant of the new key. In
other words: We have made a modulation.
Below two examples from compositions by Beethoven: |
|
Beethoven, Bagatelle for Piano op 119, No. 1
|
The first section of the form is a compound parallel period of 16 measures,
and is entirely in the home key G minor. We find an altered chord in this
section, but that is not used in a modulation: It leads (in measures 3
and 11) to a half cadence: |
|
 |
G minor:
I6 I V6
I V6eol #IVdv6 V V
2 I6 V4/3 6/5 I II6 IV6 (V6)
V |
p
p |
|
half cadence
half cadence |
|________2________|_________2_________| |_____1_____|____1____|______2_______| |
model
sequence
developing |
|_____________________________________| |___________________________________|
small sentence:
presentation
continuation
together these 8 measures form the antecedent of the period |
9 |
 |
G minor:
harmony:
see msrs. 1-6 .................................................................................
II6
IV6 I6/4 V I
|
|
HC
PAC |
|________2________|_________2_________| |_____1______|____1_____|__________2_______| |
model
sequence
developing |
|_____________________________________| |_________________________________________|
small sentence:
presentation
continuation
together these 8 measures form the consequent of the period |
|
After this first section the key changes suddenly - without transition
or modulation - to Eb major. We can consider the section from measure 17
as the B-section of the form, and observe that it is forming a contrast
with the first 16 measures of the piece. This middle section is a small
ternary:
-
Measures 17-24 are a period (thus half as long as the period in 1-16);
the antecedent ends with IAC, the consequent ends with PAC
-
Measures 25-28 form the 'middle section' of the ternary, as they contrast
with their surroundings, especially since they are all the time on the
dominant
-
The last 4 measures (29-32) are a varied repeat of 17-20, in which the
close is changed in a PAC.
|
|
17 |
 |
Eb major:
I
II6/5 V7
I 4 - 3 I6 (V6/5ovdom) IV
II6 I6/4
V7 I
|
chrom.
pass. |
|_______a______|__________b________| |_________a'________|_____________b'____________| |
antecedent
consequent |
IAC
PAC |
|
24 |
 |
Eb major :
V7 ------------------------------------------------------------
I
II6/5
|
|________________________| |___________________________| |
varied repeat |
HC |
|
After measure 32 we see a short retransition in which we modulate
back to the home key G minor, and then the recapitulation of the A-section
(from measure 37). The retransition contains a characteristic usage of
a double diminished chord in the context of an (enharmonic) modulation:
In measure 33 we could believe that the Eb chord (I in Eb major) is changed
to the dominant seventh chord Eb - G - (Bb) - Db ; In Eb major this chord
would be (V7) --> IV . As the chord resolves it becomes clear that
it is actually a double diminished triad: The tone C# - which we possibly
at first heard as Db - ascends, and Eb in the bass makes a stepwise descent.
Then it is clear that we have to interprete the 'resolution chord' (the
G minor 6/4 chord in measure 34) as I6/4 - which implies that the double
diminished triad before should be understood as Italian augmented ( #IVdv6
) in G minor: |
|
31 |
 |
|
Eb major: V7
I
I
I (V7)
forIV |
G minor:
= #IVdv6 I6/4
V(7) |
37 |
 |
G minor: I6
I V6
I V6eol #IVdv6 V |
|
Measures 33-36 are a good example of the possibility that in enharmonic
modulations (and in chromatic modulations as well) a 'diatonic connection
' between the keys can be found if we do not only look at the precise 'pivotal
moment': We could say that, in this example, the Eb chord in measure 33
is not only I in Es, but can be seen as VI in G minor as well. And then
we could call the modulation: diatonic (I becomes VI); the Italian
augmented ( #IVdv6 ) would rather be used to confirm the key of
G minor , and not so much as a means to reach that key. |
|
|
Piano
SonateaOp. 10 No. 3, second movement Largo e mesto
|
This slow movement is very complex, and I do not try to discuss it
here in its entirety. Yet a brief overview of the complete form
(please
use the complete
score as well...). At two spots in the piece we modulate using a double
diminished seventh chord: in measure 17 and in measure 56 (and these two
spots are similar).ann
The movement is a large binary, though it seems at first sight
rather a ternary:
1-29 |
A-section |
1-9 |
main theme in D minor
Periodical structure, slightly irregular: 5+4 measures. Antecedent
ends on IV! (=plagal half cadence) |
9-17 |
transition modulation from D minor to C major
It seems kind of problematic to label this section plainly, or only
as transition:
-
The character is quite 'thematic'
-
The modulation ends in C major, whereas we 'should' end in F major or
eventually A minor; only after measure 17 the 'correct key' A minor comes
into view.
|
18-29 |
B-section |
18-26 |
second theme in A minor
Initially seems not really 'thematic'. Could also be a closing section,
were it not that there is a clear closing section still to follow. |
27-29 |
closing section or codetta in A minor |
30-43 |
retransition
This section, especially at the beginning, rather gives the impression
of an independent middle section - and then we might think that we are
in a ternary form! I believe we can see it as retransition, mainly because
of the very overviewable harmony / modulation; in fact we modulate immediately
back to the home key: From the F chord, VI in A minor and III in D minor,
we reach the dominant of the home key of D minor through just a few chromatic
steps. |
44-65 |
A'-section |
44-52 |
main theme modulater from D minor to Bb major
The theme returns, varied, and becomes unstable from the sixth measure:
the consequent modulates, and ends with deceptive cadence. Caused by this
'open end' the theme is now much stronger connected with the transition. |
53-56 |
transition in Bb major
The modulation has already taken place in the main theme; the transition
confirms the key of B flat major, and he is much shorter than in measures
9-17: Of the original nine measures only the last four are quoted here. |
|
B-section |
57-65 |
second theme in D minor
Quotation (transposed to the home key) of measures 18-26 |
|
closing section
The closing section is not repeated. Instead, we move to a long: |
65-87 |
Coda in D minor
Consists of two sections: 65- 76 and 77-87. In measure 65 the beginning
of the coda overlaps with the end of the second theme. Apart from the very
complex harmony in the first section of the coda it is striking that from
measure 72 the retransition is partly quoted. |
|
|
|
 |
D minor: I
IV6/4
VII7 VII6/5 VII7 (V2) IV6 (V6/5) IV
VII4/3 |
neigbouring The IV6/4 chords
quasi subdominant
chord
on the weak beats
are passing chords |
|
 |
D minor:
(V6/5 VII7 )
I6/4 V7
I
V(6)/5
I
V
7
|
|
 |
D minor:
I(6)
V(6)/5
I
(V2) IV6dm
C major: (V2) V(6)/5
I
(V6/5)
|
|_________________________| |_______________________.......
|_____________ |
model
sequence
antecedent |
|
 |
C major:
I6/4
V7
I
(V6/5)
I6/4
V 8 ---- 7 I
(V7) ---> bVII (?)
|
A minor: = #IVdv6/5 |
____________________________|
|____________________________________| |_________ |
consequent |
IAC
PAC |
After the cadence in C major in measure 17 follows a chord that we
could hear at first as dominant seventh chord: F - A - C - Eb (We might
say: a minor seventh is added to IV in C major). But already in the next
measure it is clear that we are in the key of A minor, and that therefore
the chord consists of the notes F - A - C - Eb, and must be interpreted
as German augmented ( # IVdv6/5 ) in the relative key A minor.
ann
In measure 18 this key is reaffirmed by the repetition of the progression
with the altered chord: |
|
|
|
 |
A minor: I6/4 V #IVdv6/5
I6/4 V VII4/3 2
6/5 I6 II6/5
I6/4 V7 I
#IVdv(6)/5 |
_______| |_______________________| |
varied repeat
PAC |
|____________ |
varied repeat
of 18-21; measure
24 is added though |
|
After the cadence in A minor in measure 21 the double diminished chord
on #IV is reused, again before V - we might say as substitution of a 'normal'
subdominant. |
|
|
 |
A minor: V
VI becomes #IVdv6 V
VII4/3 2 6/5 (VII4/3 2 6/5) (VII4/3
2 6/5)
for IV for
V |
_________________________________________________________............................................... |
|
 |
(etc.....) |
|
A minor: VII4/3 I6 II6/5 V
I
replaces
V |
PAC |
____________________| |
|
 |
D minor: I
VII7 I II7
IV VII7 IV VII7 VII6/5
VII7 (V2) IV6 (V6/5) IV |
short
pedal point on I --------------------| |
IV6/4 IV6/4
as neighb. as neighb. |
|
 |
D minor: bII6
bII6 /b3 |
'minorNeapolitan'
or: "Super Neapolitan" |
Bb major: = IV6md
V7 (VII7)
VI
(V6/5) |
|_______________ |
antecedent |
In the recapitulation the main theme modulates to Bb major (see
measures 50-52), and ends 'open', unlike in the exposition: Not with PAC,
but with a deceptive cadence on VI in the new key. Probably in conjunction
therewith the transition is shortened: the first four measures are "missing".ann |
|
|
 |
Bb major: I6/4
V7
I
(V6/5) I6/4
V 8 ---- 7 I
(V7) ---> IV |
D minor: = #IVdv6/5 |
__ __________________________|
|____________________________________| |_________ |
consequent |
IAC
PAC |
|
After the cadence in C major in measure 56 follows a chord that we
could hear at first as dominant seventh chord: Bb - D - F - Ab (We might
say: a minor seventh is added to I in Bb major). But already in the next
measure it is clear that we are in the key of D minor, and that therefore
the chord consists of the notes F - A - C - Eb, and must be interpreted
as German augmented ( # IVdv6/5 ) in the home key D minor. In measure 57
this key is reaffirmed by the repetition of the progression with the altered
chord.
The difference with the exposition ann
is that we modulated there from C major to A minor (= minor third down)
and here in the recapitulation from Bb major to D minor (=major third up).
This different modulation is related to the distance between the final
chord of the preceding phrase and the altered chord (see measure 56): In
the exposition the bass moves up a fourth at the entrance of the German
augmented, whereas in the recapitulation the bass stays on the same note. |
|
|
 |
D minor: I6/4 V #IVdv6/5
I6/4 V VII4/3 2 6/5 I6
II6/5 I6/4 V7 I
#IVdv(6)/5 |
_______| |_______________________| |
varied repeat
PAC |
varied repeat
of 57-60; measure
63 is added though |
|
|
|
 |
D minor: V
VI wordt #IVdv6 V
VII4/3 2 6/5 (VII4/3 2
6/5) (VII4/3 2 6/5)
for IV
for IV |
_________________________________________________________............................................... |
|
 |
D minor: VII4/3 I6 II6/5
V I
replaces
V |
PAC |
___________________________| |