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Section E. is not at all complete yet. In the moment only this is available:  altered chords / sections 1-3, 5, and part of section 9
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1. general information
2. diminished third on #4^; 
score examples
3. enharmonization of the double diminished triad and seventh chord, and of the hard diminished seventh chord
 
score examples (choose examples from the list below):
 
Beethoven: Schubert, Winterreise D 911, Gefrorne Tr?nen
Mozart, Piano Sonata in A minor K 310   beginning of the development section of the first movement
4. #IV and IIhv in other inversions
 
score examples
5. scheme of all altered chords
6. diminished third on #4^:
score examples
7. diminished third op 7^:
score examples
8. diminished third on 2^ in minor, and on #2^ in major:
score examples
9. overview of the enharmonizations and possible modulations
10. other issues? 

 

 
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
recording by Ronald Brautigam
on fortepiano. 
The Bagatelle Op. 119 No. 1 starts at 34'35''
recording by Mikhail Pletnev (1997)
The Bagatelle Op. 119 No. 1 starts at 20'46'' (and the key of the piece is,
other than is suggested on Youtube, of course not Bb major... ;-))
recording by Gy?rgy Sebok
recording from the Fifties
recording by Rusell Sherman
recording from the Eighties
here is de complete score
of the piece (pdf)
 
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
recording by Ronald Brautigam
on fortepiano. 
The example below starts at 1:05'50''
recording by Martha Argerich (1976, live)
recording by Grigory  Sokolov (1976)
The example below starts at 10'10''
recording by Wilhelm Kempff
(1964)
here is the complete score
of the piece (pdf)
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
     ___________________________|
[continuation of this chapter]